In this paper, we consider the observation of $n$ i.i.d. mixed Poisson processes with random intensity having an unknown density $f$ on ${\mathbb R}^+$. Depending on the observation time, we propose two nonparametric adaptive strategies to estimate $f$. We use an appropriate Laguerre basis to build adaptive projection estimators and also propose kernel estimators with adaptive bandwidths. Non-asymptotic bounds of the ${\mathbb L}^2$-integrated risk are obtained in each case. The procedures are illustrated on simulated data.