No, AU does not need to be run on the client. Actually, if you run AU on the client, it is compacting or fixing the databases on the server application.
You also do not have to even have client software on the clients.
I think you misunderstood, though. What I was referring to is that in the past (it seems to have changed recently) is that you could not run the Amazing Utilities while any of the client software were open and accessing the database.