A right handed shooter will most likely be playing on the left wing.
Saying from experience, I find it easier to play on the left wing rather than the right wing [I'm right handed]. Playing on the right wing is more of a challenge to score since [most of us] are raised to use our right hand. Coming from the left wing, when the shooter takes a shot, he's facing the RIGHT side. It's just more comfortable.
Like you said about limited shots, the shooters shot options are limited when he's playing on the right wing [IMO].
Also, another tip for playing goalie is, skate/walk a FEW steps forward when the shooter is coming towards you during a breakaway/shootout. Slowly walk/skate backwards as the shooter is closing in on you, don't always stay in the net/crease.
Example:
The reason for this is because it limits the shooters shot to go top shelf, gloveside etc.