I have slightly more assumptions than the "standard" question (see e.g. this one) so the usual counterexamples that come to mind don't work (I believe).
Let $R$ be a ring and $M$ a free $R$-module. Let further $N_1$ and $N_2$ be two free direct summands. Then we have inclusions $i_j:N_j\to M$ with splittings $p_j:M\to N_j$.
Is the image of $p_2\circ i_1$ a free direct summand of $N_2$?
After I failed in producing a counter-example I thought I could maybe show that $p_2\circ i_1\circ p_1\circ i_2$ is a projector and work from there but this doesn't seem to work as well.