I am far from being an expert in this area, but I will try to present my understanding of this subject.
First of all, this is true that Hodge decomposition holds for smooth proper varieties over $\mathbf C$. However, the standard proof (that is explained below) uses the projective case as a black box.
Secondly, degeneration of Hodge-to-de Rham spectral sequence does imply that there is a filtration on $H^{n}(X,\mathbf C)$ s.t. the associated graded pieces are isomorphic to $H^{p,q}(X)$. In particular, it means that there is an abstract isomorphism of $\mathbf C$ vector spaces $H^n(X,\mathbf C)\cong \oplus_{p+q=n}H^q(X,\Omega^p_{X/\mathbf C})$. But I don't know any way to pick a canonical one just assuming degeneration of the spectral sequence.
Moreover, there is an example of a smooth compact non-Kahler complex manifold X, s.t. the associated Hodge-to-de Rham spectral sequence degenerates but the the Hodge filtration on the de Rham cohomology $H^n_{dR}(X)$ doesn't define a pure Hodge structure. An explicit situation of such a manifold is a Hopf surface $X=(\mathbf C^2 - \{0,0\})/q^{\mathbf Z}$, where $q\in \mathbf C^*$ and $|q|<1$ (action is diagonal). Then one can prove that Hodge-to-de Rham spectral sequence degenerates for $X$ but $H^1(X,\mathcal O_X)=0\neq \mathbf C=H^0(X,\Omega^1_{X/\mathbf C})$. Hence the Hodge filtration doesn't define a pure Hodge structure on $H^1_{dR}(X)$ because of the failure of hodge symmetry. Although, it doesn't imply that there is no canonical isomorphism $H^n(X,\mathbf C)\cong \oplus_{p+q=n}H^q(X,\Omega^p_{X/\mathbf C})$, it shows that it's unlikely that the 2nd statement can be a consequence of the first.
All of that being said, I will explain how to construct pure Hodge structures on cohomology of any proper smooth variety.
Let us start with $X$ being a smooth algebraic variety over $\mathbf C$. Then one can define a descending filtration on the algebraic de Rham complex $\Omega^{\bullet}_{X/\mathbf C}$ as $F^i\Omega^{\bullet}_{X/\mathbf C}=\Omega_{X/\mathbf C}^{\geq i}$. This induces a descending filtration $F^{\bullet}H^n_{dR}(X)=F^{\bullet}H^n(X,\mathbf C)$. The main result is that this construction defines a pure Hodge structure under the properness assumption on X.
Theorem 1: Let $X$ be a smooth and proper variety over $\mathbf C$, then a pair $(H^n(X,\mathbf Z), F^{\bullet}H^n(X,\mathbf C))$ is a pure Hodge structure of weight n. Moreover we have isomorphisms $F^{p}H^n(X,\mathbf C) \cap \overline{F^{n-p}H^n(X,\mathbf C)}\cong H^{q}(X,\Omega^p_{X/\mathbf C})$.
Remark 1: General theory of pure Hodge structures implies that there is a canonical isomorphism $H^n(X,\mathbf C)\cong \oplus_{p+q=n}H^{q}(X,\Omega^p_{X/\mathbf C})$.
Remark 2: Although we have a canonical decomposition $H^n(X,\mathbf C)\cong \oplus_{p+q=n}H^{q}(X,\Omega^p_{X/\mathbf C})$, a priori there is no morphism $H^q(X,\Omega^p_{X/\mathbf C}) \to H^n(X,\mathbf C)$. I don't know any purely algebraic construction of these maps, which doesn't use the fact that Hodge filtration defines a pure Hodge structure as an input. I believe this is the crucial place where theory of harmonic forms comes into play. Once you have such morphisms, Theorem $1$ is basically equivalent to the degeneration of the Hodge-to-de Rham spectral sequence.
Remark 3: I think (I may be wrong) that there is no purely algebraic proof of Theorem $1$ as well. However, what I will show below is that if one takes Theorem 1 in the projective case as an input (this is the situation where $X^{an}$ is a Kahler manifold, so one can use theory of harmonic forms), then he/she can deduce the general form of Theorem $1$ in a purely algebraic way.
Before going to explain this proof, I need to mention a lemma which eventually be the key step to reduce a case of a proper variety to the case of a projective one.
Lemma 1: Let $f:X \to Y$ be a proper birational morphism between smooth connected varieties over a field $k$, then the induced morphism $H^p(f^*):H^p(Y,\Omega^q_{Y/k}) \to H^p(X,\Omega^q_{X/k})$ is injective for any $p$ and $q$.
Proof: Recall that $H^p(f^*)$ is induced by a morphism
$$
\Omega^q_{Y/k} \to \mathbf Rf_*f^*\Omega^q_{Y/k} \to \mathbf Rf_*\Omega^q_{X/k}.
$$
Compose it with the Trace map
$$
\operatorname{Tr_f}:\mathbf Rf_*\Omega^q_{X/k} \to \Omega^q_{Y/k}.
$$
(Here we use that $\operatorname{dim}X =\operatorname{dim}Y$ because in general the Trace map is map from $:\mathbf Rf_*\Omega^q_{X/k}$ to $\Omega^{q-d}_{Y/k}[-d]$ where $d=\operatorname{dim}X -\operatorname{dim}Y$).
Now note that the formation of Trace map is compatible with restrictions to open subsets. Hence, the composition $Tr_f\circ f^*:\Omega^q_{Y/k} \to \Omega^q_{Y/k}$ is the identity morphism on an open dense subset $U\subset Y$ (because $f$ is birational and Y is normal!). Since $\Omega^q_{Y/k}$ is a locally free sheaf, we conclude that $Tr_f\circ f^*$ is also the identity morphism. Therefore, $H^p(Tr_f\circ f^*)=H^p(Tr_f)\circ H^p(f^*)$ is the identity morphism as well. So $H^p(f^*)$ is injective as desired.
Now let's come back to the proof of Theorem 1 in the proper case.
Step 1: Chow lemma and Hironaka's resolution of singularities allow us to find a smooth and projective variety $X'$ together with a map $f:X' \to X$ s.t. $f$ is birational and proper. Then there is a canonical morphisms of Hodge-to-de Rham spectral sequences for $X$ and $X'$. Namely, let
$$
E_1^{p,q}=H^q(X,\Omega^p_{X/\mathbf C}) \Rightarrow H^{p+q}_{dR}(X).
$$
$$
\downarrow{f^*}
$$
$$
E_1'^{p,q}=H^q(X',\Omega^p_{X'/\mathbf C}) \Rightarrow H^{p+q}_{dR}(X').
$$
Note that Lemma $1$ guarantees that $f^*$ is injective on the first page. Since we know that $E'$ degenerates on the first page ($d_1'^{p,q}=0$ for any $p,q$) we conclude that $d_1^{p,q}=0$ (because $f^*$ is compatible with differentials and injective). So $E_2^{p,q}=E_1^{p,q}$ and $E_2'^{p,q}=E_1'^{p,q}$, therefore $f^*$ is also injective on the second page. Then the same argument shows that $d_2^{p,q}=0$ for any $p,q$. Keep going to show that $E$ degenerates on the first page ($E_1^{p,q}=E_{\infty}^{p,q}$).
Step 2: Two consequences from the Step 1.
I) $F^pH^n_{dR}(X)\cap \overline{F^{n-p+1}H^n_{dR}(X)}=0$. We know that that his equality holds for $X'$ (b/c $X'$ is projective) and we know that Hodge filtration $F^{\bullet}H^n_{dR}(-)$ is functorial in $X$. Hence, argument in the Step $1$ says that
$$
F^pH^n_{dR}(X)\cap \overline{F^{n-p+1}H^n_{dR}(X)} \subset F^pH^n_{dR}(X')\cap \overline{F^{n-p+1}H^n_{dR}(X')}=0.
$$
II) We have a canonical isomorphism $F^pH^n_{dR}(X)/F^{p+1}H^n_{dR}(X)\cong H^{q}(X,\Omega^p_{X/\mathbf C})$. This is a direct consequence of the degeneration of the Hodge-to-de Rham spectral sequence.
Step 3: The last thing we need to check is that $F^pH^n_{dR}(X)+\overline{F^{n-p+1}H^n_{dR}(X)}=H^n_{dR}(X)$.
Since $F^pH^n_{dR}(X)$ and $\overline{F^{n-p+1}H^n_{dR}(X)}$ are disjoint inside $H^n_{dR}(X)$, it suffices to prove the equality
$$
\operatorname{dim}(F^pH^n_{dR}(X))+\operatorname{dim}(\overline{F^{n-p+1}H^n_{dR}(X)})=\operatorname{dim}(H^n_{dR}(X)).
$$
Ok, Consequence II from Step $2$ implies that
$$
\operatorname{dim}(F^pH^n_{dR}(X))=\sum_{i\geq p}h^{i,n-i},
$$
$$\operatorname{dim}(\overline{F^{n-p+1}H^n_{dR}(X)} = \sum_{i\geq n-p+1}h^{i,n-i},
$$
$$
\operatorname{dim}(H^n_{dR}(X))=\sum_{i\geq 0}h^{i,n-i} \text{, where } h^{p,q}= \operatorname{dim}H^q(X,\Omega^p_{X/\mathbf C}).
$$
Since we know that $F^pH^n_{dR}(X)$ and $\overline{F^{n-p+1}}H^n_{dR}(X)$ are disjoint we conclude that
$$
\sum_{i\geq p}h^{i,n-i} +\sum_{i\geq n-p+1}h^{i,n-i} \leq \sum_{i\geq 0}h^{i,n-i}.
$$
Subtract $\sum_{i\geq n-p+1}h^{i,n-i}$ to obtain an inequality
$$
\sum_{i\geq p}h^{i,n-i} \leq \sum_{i\leq n-p}h^{i,n-i} (*)
$$
Our goal is to prove that this is actually an equality. Here comes the miracle! We use an additional symmetry between hodge numbers of interchange left and right hand sides of this inequality to actually show that it is an equality! Namely, we apply Serre duality to see that $h^{p,q}=h^{d-p,d-q}$ where $d=\operatorname{dim} X$. Then we can rewrite both hand sides of $(*)$ in a different way.
$$
\sum_{i\geq p}h^{i,n-i}=\sum_{i\geq p}h^{d-i,d-n+i}=\sum_{j\leq d-p=(2d-n)-(d-n+p)} h^{j,(2d-n)-j},
$$
$$
\sum_{i\leq n-p}h^{i,n-i}=\sum_{i\leq n-p}h^{d-i,d-n+i}=\sum_{j\geq d-n+p}h^{j,2d-n-j}.
$$
But now observe that inequality (*) (applied to $n'=2d-n,p'=N-n+p$) guarantees that
$$
\sum_{i\leq n-p}h^{i,n-i}=\sum_{j\geq d-n+p}h^{j,2d-n-j} \leq \sum_{j\geq d-n+p}h^{j,2d-n-j} =\sum_{i\leq n-p}h^{i,n-i}.
$$
Therefore $\sum_{i\geq p}h^{i,n-i}=\sum_{i\leq n-p}h^{i,n-i}$! And we showed that this is equivalent to the equality $$
F^pH^n_{dR}(X)+\overline{F^{n-p+1}H^n_{dR}(X)}=H^n_{dR}(X).
$$
In other words we showed that $(H^n(X,\mathbf Z), F^{\bullet}H^n(X,\mathbf C))$ is a pure Hodge structure of weight $d$.
Best Answer
[I misunderstood Torsten Ekedahl's earlier comment. I'm reverting the lemma to its original form which was a bit stronger.]
Since the question seemed to resonate with me, I've been thinking about this on and off (but mostly off) for a couple of days now. Here's what I've come up with.
What seems to make the example of complete intersections work is the fact that the Hodge numbers can be computed by formulas independent of the characteristic (a standard generating function can be found in SGA7, exp XI). Here's the underlying principle.
Proof. It degenerates on $\mathcal{X_\eta}$ by Hodge theory. This plus semicontinuity implies $$\dim E_1(\mathcal{X}_0)\ge \dim E_\infty(\mathcal{X}_0)\ge \dim E_\infty(\mathcal{X}_\eta) =\dim E_1(\mathcal{X}_\eta)=\dim E_1(\mathcal{X}_0)$$
This can be used to check degeneration for the following cases:
Ex 1. Complete intersections in projective spaces as noted already.
Ex 2. Products of smooth projective curves and complete intersections. Use Kuenneth and the fact that curves lift into characteristic $0$ (the obstruction lies in $H^2$ of the tangent sheaf; or see Oort, Compositio 1971).
Ex 3. Certain cyclic branched covers of projective space, and more generally certain hypersurfaces in weighted projective space. I'm too lazy to say what "certain" means exactly. But a careful reading of Dolgachev's notes on weighted projective spaces ought to yield something more precise.