What can we say about two immersed submanifolds if their intersection is open in one of them?

Question

Suppose $N_1,N_2\subseteq M$ are two immersed submanifolds of $M$,and $i_1:N_1\rightarrow M$,$ i_2:N_2\rightarrow M$ are the inclusion maps respectivly.Suppose $i_1(N_1)\cap i_2(N_2) \subseteq M$ is an open subset of $i_1(N_1)$, Then for any $p\in i_1(N_1)\cap i_2(N_2)$, is the statment $T_pi_1(N_1)\subseteq T_pi_2(N_2)$ right?
The original qustion I wonder is that if the intersection of two submanifold $N_1\cap N_2$ is an open set in one of them, w.l.o.g in $N_1$, then is the intersection a submanifold of the other one, i.e. is $N_1 \cap N_2$ a submanifold of $N_2$?
Actually ,this question rise from the foliation geometry. It seems that I'm lost in some kind of routine arguments about immersed submanifold/regular submanifold.