No, you can't use the same global variable on two different screens. Why even use player1 or player2 variables? In the when StartButton click block you could just update the TinyWebDB1 tags to P1.Text and P2.Text directly. No need for the middleman. And then in the when TinyWebDB1 gotvalue block, are you trying to set p1 and p2 variables to the value of TinyWebDB? If so, then there's no need, unless you are using those variables in some other part of you app, in which case those instances could be replaced with TinyWebDB1's tags and values. From what I've gathered, you have two screens (NameEntry and GameScreen) so you could call TinyWebDB1's values in gamescreen, where ever you used getP1name or getP2name. I'm pretty new at MIT appinventor so if I got something wrong then sorry.
You should use TinyDB, not TinyWebDb. TinyDB data stays on the device and will be much faster than TinyWebDB. It will also work when you are offline. Also, unless you host your own TinyWebDB instance, anyone can access the data you put into TinyWebDB.