I don't think they will, at least not in their default configuration. They do have the capability to communicate between the satellites but I'm not sure that they are setup to deliver data going strictly between two Starlink'd users without going through a ground station. There's also the problem of just keeping them up, being low flying satellites they'll need periodic boosts which are planned from the ground iirc.

Both of these could be turned over to an automated system though I suppose.