Can two people with different one time pads securely exchange a message like this?
The general scheme is called Three-pass protocol and works for all commutative ciphers. It is secure for some of them, but xor (and modular addition) are insecure choices.

Your scheme:

A->B: $c_1 = m \oplus a$

B->A: $c_2 = c_1 \oplus b$

A->B: $c_3 = c_2 \oplus a$

B computes $m = c_3 \oplus b$

an attacker sees all of $c_1$, $c_2$ and $c_3$. So they can compute:

$c_1 \oplus c_2 \oplus c_3 = (m \oplus a) \oplus c_2 \oplus (c_2 \oplus a) = m$

and thus recovers the message.