So basically you go up to the teller and just ask for a deposit slip and fill everything out according to the account name/number and that sort of thing?
Actually it was simpler than that. I had written the deposit details (name/institution/acct #) on a post-in and handed it to the teller and told her I wanted to make a deposit to that account. She fills out the deposit slip. Typically they have to look in their system for the account, or in the case of Credit Union, find the receiving CU. One teller asked me 'where,' as in what city/state was the bank and I just smiled at her and said 'I don't know.' She just smiled and said 'ok.' She couldn't care less why. Again my transaction was less than US$100 so I think these types of transactions have gotten quite common.
I also learned that you can deposit to any Credit Union in the US from any other Credit Union in the US. I did that a couple times. Once I miss-wrote (scrolled it wrong) the name of some obscure CU and the teller and I spend about 4 minutes guessing what it might be before we figured it out. Again, she couldn't care less that I didn't know the recipient, nor what the funds were for. That transaction was probably in the $100-$400 range.
So Bitquick stands between, i.e. escrows the transactions, alleviating security concerns. If Localbitcoins does that as well, my sole attention can be directed at transaction costs, which are significant.