Here's a bivariate Poisson whose marginals are Poisson

http://en.wikipedia.org/wiki/Poisson...n_distribution

From what I remember, there's not just "one" bivariate Poisson, but several people have developed several different bivariate (and trivariate) distributions that have the appropriate properties (I think some of them are listed in the "See Other" of that wikipedia page)

Multivariate probability distributions are a weird thing; I think the Poisson is a good example -- there's not always a clear "multivariate version" of each univariate distribution, but people can develop multivariate distributions that have certain properties that resemble a univariate distribution, but these may not be unique (as in the multivariate Poisson cases)