Here is a solution sent to me by Ryan Batterman:
Let n be the number of friends in the group. Rip the paper into n
pieces, and have the friends stand in a circle. The first friend will
write a number which is a random number + his salary on his piece of
paper and hand it off to the person in front of him. Then the second
person will add his salary to the number, writing it on his clean
piece of paper; he hands this off to the 3rd person. This cycle
continues until the first person (who originally wrote the
pseudorandom number on his sheet) has a sheet representing the sum of
the salaries and the random number. He subtracts the random number,
and tells everyone the sum of the salaries.