The players, not the umpire, decide who serves and receives first.
That's why you'll see them pointing at themselves or their partners after the coin toss. Also the winner of the coin toss decides which side they'll start on... which is very important depending on the drift of the stadium.
Indeed the partners decide who should serve first and the umpire will take note of this in his score sheet.
One thing to note as a pair is to remember which one of you has served before "service over". Then when the service comes back to you, the partner who has not served will be the next to serve. Service alternates between the two partners.