 Solution

Reordering as $A_1{A}_2{B}_1{B}_2$ gives ${\alpha }^2|0000⟩+\alpha \beta (|0011⟩+|1100⟩)+{\beta }^2|1111⟩$. Each party applies CNOT $1\to 2$ and measures qubit 2. When both obtain 1, qubits $A_1{B}_1$ are proportional to $|00⟩+|11⟩$. The success probability is $2|\alpha \beta {|}^2$.