I had to go through it last summer to pull some money out.
I've been verified for a while so I don't recall the exact amount but you were able to deposit and withdraw small sums without previously being verified. It appears now you will need to be verified to do any deposits or withdrawals.
Verification was mandatory to use the Dwolla deposit option for at least the past year, not sure about the other methods.
I know when I started using MtGox (via Dwolla) there was no verification process from either company but that was back in 2011. I do recall needing to be verified by both late 2012 or early 2013. This makes me wonder though, if everyone was verified by BOTH sides, why was the Dwolla account frozen? I understand they didn't have the licenses but they should be able to provide ample evidence that they did follow KYC and AML laws. Makes me wonder if the DHS will eventually release those funds.