I believe ESRI would be the best people to talk to about this. I've setup Collector a few different times and it's a really cool product, especially when using feature services directly from ArcGIS Server, but there are a lot of different factors that could be causing you problems. Without having your map or data it's impossible to tell you what you're doing wrong.
One thing you may be missing is that when you first start the app and login, you need to enter the URL of your organization, for example:
organization.maps.arcgis.com
And then enter your username/password that is associated with the organization. This username also needs to be set to either an Administrator or Publisher role.
This ESRI post seems to be exactly what you're asking:
http://forums.arcgis.com/threads/89386-Please-use-an-organizational-account-error-on-collector
kylie from ESRI:
There are two different accounts that you can get for no cost. One is
a public account. Public accounts are not supported in Collector. The
other is a 30-day free trial of an ArcGIS subscription. Those are
supported with Collector. You will need to set up your organization,
and then you will be able to log in with the account tied to that
30-day trial. By set up your organization, it is not paying for it.
When you create the free trial you will get email that explains what
you need to do.
What's all in your map?
Are you adding a shapefile? Did you make sure to check "Publish this file as a feature layer"? The editable layer has to be a feature service. Also, you need to turn editing on for the feature. I'll add some screenshots.
If editing says Disabled, you need to click Edit and at the bottom of the page turn it on.
Go into the map on AGO and make sure you can edit it there.
AGOL web maps are not restricted to just hosted feature services inside your organization. You can use ArcGIS Server feature services but sometimes depending on your server network you may have a message displayed in the web map that editing is disabled due to the server coming from an internal network. This is only a limitation in web map in ArcGIS Online and Collector will still be able to edit these services. Also with the web map saying these layers are not editable you will be required to have an editable layer in the web map like a dummy hosted feature service that will give the web map the Collector flag which helps the app find maps that can be used in Collector.
Best Answer
You only need to enabled Feature Access on the service when publishing and have some variant of Add/Update/Delete enabled.
Also you need to share the map and layers to a group that your other org members are a part off. The app will only display maps shared across groups that you are a member of or maps that exist under your My Content. These maps also need to have an editable layer included to be picked up by the app when it accesses your available content.