Our experience was they made the offer over the phone (with son writing as fast as he could), then followed up with an e-mail. The offer was broken down into various components. No time limit was given initially but my son told the school he had other visits scheduled and asked if he would be able to complete those before deciding. The coach was agreeable and asked that he decide within 3 days after the last scheduled visit.
We weren't in a July 1 situation, more like late September, so it may differ for the red hot highly sought D-1 prospects.
The phone doesn't necessarily start ringing off the hook on July 1 except for the really highly and widely sought after guys, so no players should be disappointed if it doesn't ring that day. My son didn't start getting calls until mid-August, but eventually had around 10 schools call.