It seems that this always boils down to a single solution... that one side claims the other side wants.
"Deport everyone" - that one really irritates me. I have not seen, nor heard anyone, anywhere, advocate mass deportation.
"Amnesty for all illegals" - Even the far left doesn't give voice to that, even if it is what they desire.
There is no single solution. But, it's not rocket science either. As Bigun says, turn off the magnet. Make the punishment severe for hiring illegals. Erect some sort of fence. Dry up the benefits to illegals. Most of those that are here, are here. We can debate what criteria we can use to qualify for citizenship, or benefits, or some form of purgatory, but those that are here aren't going anywhere.