I'd like to add a field where users can add their own tags, but also select from tags created by other users through a picklist. I believe there used to be an option on the Multi-choice field that supported something like this, but I can't seem to find it.
Has any done this, or does anyone have suggestions for how this might be accomplished?
Your are correct that this can be done thru the Multi-Choice Field.
Create the field like normal in your table and set up any values you want to begin with.
On the Form the User would be using, if you click the 'Edit this Input' option - you will have the option to check for 'Allow users to add new options'.
You would need to do this on any form that you want the user to be able to add additional options on.