Does anyone here permit their babies to watch while eating?

I assume you mean watching the television or some type of screen. If that's the case, then yes. My daughter watches TV while she eats, but in all honesty, she doesn't mind it all too much. She's more fixated on eating her food rather than actually absorbing what's happening on screen. Usually, once she's done eating she wants to tune in to what's going on on the TV.