Find the smallest summed distances by uniquely pairing elements of one set to elements of another set
This is the problem of finding the maximum matching in a weighted bipartite graph.  There are efficient algorithms that solve this problem in polynomial time.

Basically, you have a complete bipartite graph with $2m$ vertices.  $S$ forms one set of vertices; $T$ forms the other set of vertices.  For each $i,j$, create an edge from $s_i$ to $t_j$ whose weight is equal to $-d(s_i,t_j)$, the negation of the distance between $s_i$ and $t_j$.

Now, you want to find a matching whose weight is as large as possible.  This corresponds to a set of pairs $(s_i,t_j)$ whose total distance is as small as possible.  From there, you can use existing algorithms on this graph to find the maximum-weight matching.