Do left handed people feel odd shaking hands with their right?

If I meet someone left handed would it make sense for both parties to shake with their dominant hand?

How’s this work? I’m sure all along I’ve just shook hands with lefties, and they simply used their right hand.

If so, they must sometimes meet lefties too and then maybe these two lefties shake with their non-dominant hand. A double awkwardness.

I bet there’s some who have experiences with this…