It should be written with parentheses to avoid ambiguity, yes. If you think about the "sum" symbol as a function, it makes sense:
$$
\sum(\cdot)
$$
This is a function which takes a list $\{x_1,x_2,\ldots\}$ of numbers (or other mathematical objects which you might want to add), and adds them in order. This list could be finite or infinite: the "sum" function figures out how long the list is, and adjusts its indexing accordingly (i.e. if there are 10 things in the list, your index will go from $1$ to $10$).
If you want to add a sequence which is itself the addition of two sequences, like your example of $\{x_1+y_1,x_2+y_2\}$, you'll need to drop the whole sequence into the function:
$$
\sum(\{x_1+y_1,x_2+y_2\})=\sum_{i=1}^2(x_i+y_i)
$$
For finite sums of numbers, we always have the property that
$$
\sum(\{x_1,x_2,\ldots\}+\{y_1,y_2,\ldots\})=\sum(\{x_1,x_2,\ldots\})+\sum(\{y_1,y_2,\ldots\})
$$ however the following would have a different interpretation:
$$
\sum(\{x_1,x_2,\ldots\})+\{y_1,y_2,\ldots\}
$$
Hence the parentheses!
Interesting, yet more advanced side note: "breaking up" a sum doesn't always work if the sequences are infinite!
Not everyone was taught what you say. I was not, for example. I was never taught how to write expressions with exponents in-line, so I never found out what the canonic meaning of x^a+b actually is.
What I was taught is that whenever there is some confusion and there may exist two ways of interpreting an expression, I should use parentheses. And that is exactly what you need to start doing.
The thing is that by now, the notation has become so widely used with no central rule telling us what the only proper way of evaluation is, that it no longer makes much sense to try to impose a world-wide standard.
Taking this into consideration, the answers are:
- There is no general correct rule for this kind of operation. Brackets are the way to go. There is no international standart.
- The following expressions should be evaluated as "input unclear". If you get an expression like that to evaluate, ask the author of the expression to further explain what they meant.
Best Answer
You can't get it wrong with a multiplication on the left, the sum is taken as a whole and clearly
$$a\sum b_i=a\left(\sum b_i\right).$$
Malicious guy could tell you that there is ambiguity in
$$\sum a_i\ b$$ which could be interpreted as
$$\left(\sum a_i\right)b$$
or $$\sum \left(a_i\ b\right)$$
but by the distributivity law, these are equivalent.
Different is
$$\sum a_i+b$$ which could be understood as
$$\left(\sum a_i\right)+b$$ or $$\sum\left(a_i+b\right).$$
Without parenthesis, the first interpretation holds.