At right are solutions for the T- and Z-tetrominos. Below is a solution for an octomino.
It's pretty obvious that any solution to this problem for n distinct polyominos has to cover 2n-1 squares, and the individual polyominos have to have 2n-1 squares. I will leave to someone else the discovery of a 16-omino, of which five copies can be combined to produce all 31 possible overlaps of one square each. Let me know.