Both players play their own ball on each hole.  The lowest net score of the 2 players is recorded as the team score for each hole.  The net score for a hole is determined by subtracting any strokes that the player receives (based on handicap) from the players actual score.   
How do I know what holes I get strokes on? (Click Here).

Handicap allowance is 90% of Course Handicap.  If a team has a handicap difference of more than eight strokes then 80% of course handicaps are to be used.
 The above is per USGA handicap Sec 9-4(b)(ii).  

Each player should record all of their actual scores on every hole on the score card.