I can see mine on my Droid just fine (just checked) - version 1.9.0. Clear you apps cache, and/or data. Or uninstall/reinstall it.
Might be something to take up on their forums if that doesn't work - Tapatalk Community Forum
Edit - There is a bug there.
When I signed into tapatalk and clicked on this forum from my favorites, then logged in to it, I could see the PM's tab on the top bar there and could get to my pm's. But if I hit the back button and went back to the screen showing favorites, then clicked on this forum again, the PM tab is gone from the top bar. Even though you're still logged in.
Had to log out and back in again to see the PM tab again. So seems there is a bug there in the app.