Any user shall be able to assign any user to an issue

[Jennifer Schwanenberg]

I want to be able to assign any user to an issue at any point of the process.

Right now I am only able to assign myself or "automated", but automated falls back to "unassigned".

[John Funk]

Hi Jennifer - Welcome to the Atlassian Community!

Since you are on a Free account, you will not be able to modify the Permissions. 

But take a look at the Permission Scheme for the project, and it should say Application access (Andy logged in user) for both the Assignable User and Assign Issues permissions. 

Does it say that? 

[Chris Buzon]

This is going to potentially be a bit confusing to explain, so if you're the admin I'd suggest using the permissions helper to find the problem(s)

Once in any project press "." (period) for the admin shortcuts and look for "permissions".  You want the permissions helper shortcut (you can find that in System settings too).

Select a user other than yourself, then an issue you want them to be assignable to (it's doing some diagnostics here), then scroll down to Assignable User (and you'll do this again to check "assign issues") - it will show you what's wrong in your settings.

Basically your users don't have one or more permissions set up for the group they're in. There can be several more specific problems, like the permissions scheme on the project being the culprit, or the project roles being set up incorrectly for what you need.

[Chris Buzon]