"Hello, I was wondering whether there is a way for me to login as user2, for example, while running a process, like mozilla, as user1. Is there such a possibility?"
Yes. You can use virtual consoles to log in as other users, including root. The number of virtual consoles varies from one distribution to another. You may have 8 virtual consoles at <Ctrl> <Alt> <F1> through <Ctrl> <Alt> <F8>
Also a few Linux systems use <Alt> <Fx> for virtual consoles instead of <Ctrl> <Alt> <F3>
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Steve Stites