Here's my best shot at it:

An individual player can expect a pocket pair every 1 in 17 hands and upon playing that PP will flop a set or better 11.75% of the time. If you play every pocket pair, you'll flop a set 0.69% (11.75%/17) of the time (once every 145 hands or so).

For two players, the probability of flopping a set is the sum of their individual probabilities less the probability of both flopping a set on the same hand:

P[A U B] = P[A]+ P[B] - P[AB]

Unfortunately, the two events aren’t independent, so there are a few hoops to go through to calculate AB. First, assume that player A has a pocket pair (1/17). There are 72 ways the second opponent can be dealt a pocket pair that can yield a set (combin[4,2]*12) out of 1225 hands (combin[50,2]). There are 184 (2*2*46) ways to flop each player a set or better out of 17296 (combin[48,3]) possible flops. This gives P[AB] = P[AB|A]P[A] = (72/1225)*(184/17296)*(1/17) = 3.678x10^(-5). Thus,

P[A U B] = 0.006915 + 0.006915 – 0.00003678 = 0.013793

This gets a little more difficult as the number of players increases. For example, for three players:

P[A U B U C] = P[A] + P[B] + P[C] – P[AB] – P[AC] – P[BC] + P[ABC]

We already computed P[AB] (= P[AC] = P[BC]), but to get P[ABC], we need to assume that all three players have PP’s of different rank (probability is (1/17)*(72/1225)*(66/1128)) and that one of the 6 (2*2*2) three set flops occurs out of 17296 possible flops. Thus P[ABC] = (1/17)*(72/1225)*(66/1128)*(6/17296) = 7.02x10^(-8). So,

P[A U B U C] = 3*(0.006915) – 3*(0.00003678) + 0.0000000702 = 0.020634.

For more than three players, you will encounter terms for the probability of four (or more) players flopping a set, which is not possible. As a result, the formula reduces to:

P[at least one of N players hitting a set on the flop] = combin(N,1)*(0.006915) – combin(N,2)*(0.00003678) + combin(N,3)*(0.0000000702).

Here is a table summarizing the results:

Players...................Probability of a set
2..............................0.013793
3..............................0.020634
4..............................0.027373
5..............................0.034207
6..............................0.040938
7..............................0.047633
8..............................0.054292
9..............................0.060915
10............................0.067501