It depends. At one time our library was cheating and using a personal Ancestry account. During that time you had to use one of two computers in the LH&G department because the LH&G librarians would have to log you onto the account. They got caught and had to pay for Ancestry Library Edition. Since they went to Library Edition any computer on their network (at any branch) will work since it validates through the main IP address. There was a similar situation at the next county over. I could access Ancestry at the nearest branch (about 10 miles away) but if I asked about it, I'd be told I'd have to drive to the main branch and use one of the computers in the LH&G department.
P.S. Some public computers are set up to restrict access to Explorer (directory viewer). In this case you may have to play around with available apps to get an explorer window to run RM. Also, the launcher (RootsMagic Launcher.exe) at the root of the thumb drive's file system may not work. You may have to go into the RootsMagic directory and launch the program by clicking RootsMagic.exe.
P.P.S. If they are running Ancestry Library Edition, you can take a laptop, access their WiFi and run RootsMagic and Ancestry on your laptop.