If I create a body mover (like BodyVelocity) and apply it to a player in a local script. Is the change visible to all players?
Yes, other players will see the player moving from the BodyVelocity but they will not be able to see the instance itself. 
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Oh, also if I create the BodyVelocity instance in the localscript will the effect be much faster than making the instance in serverscript?
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Yes, it would be there instantly if it is created on the client. If it is created on the server there may be a small delay.
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