The mountain rescue here in BC are volunteers (http://www.northshorerescue.com/) - I think their main problem is turning down new applicants rather than finding them.

The neighboring region had a recent tragedy. There the police (RCMP=state troopers) are responsible for rescues but aren't experts. A pilot reported an SOS in the snow, the police followed their procedure, which was to check for missing person reports. Since nobody was missing they ignored the SOS.
When a second SOS was seen a few days later a volunteer search was mounted and found one survivor - a lost cross country skier, unfortunately his wife had died of exposure.

In the UK the marine rescue services (RNLI) are all volunteer crewed and funded. This is because they date from before there was much government but they are very defensive of their independent status. It lets them use the best equipment, procedures and people free from political interference.