One of the oldest dilemmas in the world - when you really want to talk to someone but there's no one there to listen. And yes, it's very likely that Richard, being human just like the rest of us, had that problem from time to time.
It's probably difficult to realize that because for the most part, let's face it, we're only seeing Richard's right arm, in the guise of a zany, lovable character who has the audience's undivided attention (and that's a good thing, mind you). I think the closest we've come to seeing a vulnerable side of Richard was during Jim's memorial, and in the interviews done afterward.
But there was a very telling line that Richard read from a fan letter at the memorial: "I mean, things change and all, but the Muppets - they were always there." We've read about Richard hanging around backstage at the studios in London, carrying on a two-way chat with a puppet on his arm. When the humans weren't able/willing to listen, Richard always had Scooter and a whole crew of friendly Muppets to chat with.
So the Muppet crew had a few advantages over others - they could use the Muppets to act out their own quirks and frustrations, and they always had someone to talk to. Both are lot more fun, and a heck of a lot cheaper, than a visit to the shrink.