I was going to ask this same question. I noticed that there are some anecdotal answers available here, but I wanted to see some documentation. Here is what I've found.
Based on Android Open Source Project documentation it seems that applications do run in the background. It says this:
Secondary. Any user added to the device other than the primary user.
Secondary users can be removed (either by themselves or by the primary
user) and cannot impact other users on a device. These users can run
in the background and continue to have network connectivity.
(Archived page: Supporting Multiple Users | Android Open Source Project (User Types))
However, if you're running a version of android built for automotive use, it's a different story:
The following exceptions apply to headless system user and regular
(secondary) Users in Automotive:
- The headless system user does not support work profiles.
- By default, regular (secondary) users have full access to phone calls and texts.
- By default, regular (secondary) users do not run in the background.
(Archived page: Supporting Multiple Users | Android Open Source Project (Android Automotive Multi-User - Caveats))
Personally I would like the ability to choose whether a user is running or not, Android seems to have the ability to modify this, but unfortunately it's built into the "flavor" of Android that you're using. Perhaps there are some custom roms out there that have modified this.