Let Wk be the time to the kth birth in a pure birth process starting from X(0) = 0. Can you establish the equivalence?
Pr(W1 > t, W2 > t + s) = P0(t)[P0(s) + P1(s)]?
Can you determine the joint density for W1 and W2, and then the joint density of S0 = W1 and S1 = W2 – W1 ?