Let $i_K^W:K^k\to W^w$, $i_N^W:N^n\to W^w$ be smooth embeddings of smooth manifolds with boundary and assume that $N$ resp. $K$ meet $\partial W$ transversely in $\partial N$ resp. $\partial K$.
11.8 Definition. In the above situation, denote $i_N^W[N]\in H_n(W,\partial W)$ by $[N]_W$. Also define $\tau_N^W=D_W([N]_W)\in H^{w-n}(W)$, the Thom class. Here $D_W:H_n(W,\partial W) \overset{\approx}\to H^w-n(W)$.
11.9 Theorem. With the assumptions above, including that $K \cap\kern-0.7em|\kern0.7em N$ in $W$, we have $$\tau^W_{K\cap N}=\tau^W_{K}\cup \tau^W_{ N}$$ and equivalently, $$[K\cap N]_W = [N]_W \bullet [K]_W.$$
Do we really need the transversal intersections on the boundarys? Can't we just proceed as follows?
We consider manifolds $N,K$ with non empty boundarys $\partial N,\, \partial K \subset \partial W$, intersecting transversally outside their boundarys, not necessary transversally on the boundaries.
By defining $$N' = N - \partial N,\quad K'=K-\partial K,$$we get two smooth manifolds meeting $\partial W$ transversely in $\partial N= \emptyset = \partial K$. Since their boundarys lie in $\partial W$, we have $$[N]_W=[N']_W,\quad [K]_W=[K']_W \quad\text{and}\quad [K\cap N]_W=[K'\cap N']_W.$$These manifolds $N',K'$ obviously meet the conditions needed for the Theorem. Therefore $$[K\cap N]_W =[K'\cap N']_W = [N']_W \bullet [K']_W= [N]_W \bullet [K]_W.$$
Noncompact manifolds do not have fundamental classes. There is no element named $[N']_W$ satisfying the properties you want.