If someone wants to cash an on-us check, then funds are verified, but if an on-us check is being deposited by another customer, we accept the deposit and let the system determine at posting if funds are available. If there are not funds at time of posting, depending on customer relationship, we'd either pay into OD or charge back to the depositor. We'd never be able to verify all on-us checks in deposits, especially our business deposits.
If it was an extremely large check in a one-off deposit, we'd probably look at the maker's account and if funds were not available, we'd try to contact the maker to see if they were expecting a wire or some other deposit coming in to cover the check.