I would do it this way: 1) find the position on the elements equal to $1$ on the diagonal, 2) select the corresponding rows and columns.
operate[mat_] := With[{pos = Flatten@Position[Diagonal@mat, 1]}, mat[[pos, pos]]]
operate@{{1, 0}, {1, 0}}
(* {{1}} *)
operate@{{1, 0, 1}, {1, 0, 1}, {0, 0, 1}}
(* {{1, 1}, {0, 1}} *)
operate@{{0, 0, 1, 1}, {1, 1, 0, 1}, {0, 1, 1, 0}, {1, 0, 1, 0}}
(* {{1, 0}, {1, 1}} *)
operate/@list
(* {{}, {{1, 1, 1}, {0, 1, 0}, {1, 1, 1}}, {}, {}, {}, {}, {}, {}, {}} *)
Edit The question has been changed (replacing "diagonal" with row) so of course this answer no longer makes sense...