This is not true in Ohio. I was laid off from a public school district in May 2008. My last day of work was 6/6 and I started collecting unemployment the following week. I was still collecting paychecks for my job with the school but that was for time already worked. They just spread the pay out over 12 months instead of 9 months. It does not impact your unemployment benefits.
Every state has different unemployment rules and since the OP's friends were having difficulty, this is most likely why. That doesn't mean it is not true in THEIR case.