Arm chair flyer;
I don't filter any of them, and they are all available for all crew to read they come on a pad with individual numbers printed on them so each one can be traced. All of our CC go on he same CRM courses as the pilots, and so do our ground ops staff. There's a good chance if there is a problem we know at the front, but we need a concise accurate report of what the CC member is seeing, when possible they get to ride on a sim jump seat to see how we handle emergencies at the front (it took me a long time to get that approved, but its been really worthwhile).
Any member of crew can file an air safety report, but I prefer CC to discuss it with the pilots before filing so that it is based on technical fact rather than a supposition, I have read some fairly inaccurate ASRs based on the fact that the crew member thought something was wrong and that the pilots failed to inform her, in one case there was, it was an autopilot runaway and the pilots had their hands too full, just before DA to tell the CC what was happening. They sorted the problem, landed went down the pub and had a laugh about it, the CC filed an ASR believing the FD had mis-handled the aircraft (empty positioning sector). OK they should have told her, but maybe she should have hung around and asked what went on rather than going straight into print. Their MOR and the snag sheet arrived with me at the same time as her ASR, a well done for good flying and a wrist slap for poor CRM went to the pilots, I then had to explain to the CC member with the P1 what it all meant, and I have few enough hours in my day as it is.
I've also known a CC member who called the front during initial climb to report "a buzz and vibration" from her seat nothing showed up up front until after landing the crew found a section of rudder trim tab had broken off as they climbed. Her concerns were very valid and the crew spent a long time analysing things, turned out afterwards the buzz started as the gear retracted and the pilots didn't notice over the noise of the nose gear coming up. According to the P1 her report was as accurate as hell, but the problem left them shaking their heads in tracing it.
I value the CC, but I value accurate reporting as well.