this language and acceptance by queue automaton
Working with a queue automaton for me is intuitively equivalent to working with a circular tape.
Read the input $w$ from the automaton, and put it on the tape. Add two markers $A$ and $B$, so we obtain $Aw_1Bw_2$. Now we can one by one eat a letter after $A$ and one after $B$.
For the deterministic variant it suffices to put $B$ exactly in the middle of the input string. That position can be found using a method similar to this: How to find middle element of linked list in one pass?