This is a great topic because I think it is very relevant to the running of SAR ops. As I've previously stated, I've coordinated SAR responses for missing bushwalkers and I can tell everyone right now that if you're legally able to fly, have a suitable aircraft and on the ground and willing to give your time etc and listen to the coordinator, you will be very involved in the incident.
My experience involved two missing bushwalkers in the Carnarvon Gorge, overdue two nights, near freezing weather on the top of the ridge line they were supposed to be traversing and a rescue choper that was on it's way and then diverted to a cardiac case on an island off Rockhampton, with a further 4 hr delay. After jumping up and down a bit, I finally managed to convince my powers that be that we needed local assets straight away due to enclement weather on the way and the situation not looking good for the missing two. Within an hour I had two local choppers (R22 and R44) at the Bandana Airstrip near the Gorge, ready to go. One of the pilots is a pretty good mate of mine and both are only Commercial Licenced.
As Destiny said, AUSSAR coordinate a search and I tell you that once they are on board, you don't have any two and fro-ing about who is funding the whole thing. They do, simple. That saves so many managers of other agencies from having to make decisions based on money etc because it all comes from the central pool. Also, as Turbo stated, go and see your local police and tell them what you can do.
Oh yeah, we ended up finding the two missing walkers after two days of searching, using 4 choppers, a cessna and a Dornier from Cairns not to mention the numerous SES volunteers (gotta love those carrots!). One was perched on the edge of a cliff and the other was stuck on a ledge with an 80 metre drop off to the Gorge below.