The only other company I know of that doesn't allow F/O's to do everything skip would do on his handling sector takeoff is Crossair/Swiss.
In our company, the only distinction between Skip an Effo during takeoff is that it remains the Captain's decision to abort, so once Effo sets takeoff thrust, the Captain will place his hand on the thrust levers until V1. Other than that there are no differences.
Some airlines even allow the F/O to decide whether or not to abort, and therefore he keeps his hand on the thrust levers until V1.
It worries me a bit that some outfits don't allow their Effo's to complete most of the landing and/or take off. Could be a bit of an obstacle when said Effo comes up for Command!
Flying Clog
bmi Regional, UK