In the past members of the general public were on the grant committees (chosen randomly by deed poll, but then interviewed, which evidently carries an element of bias), so there has been some level of public involvement. i'm not sure if that still happens today. The other seats on those committees are "open", in the fact that thye're advertised and people apply to be on them.
People's Millions was two stage application process - any charity / vol org / SE / LA etc was able to apply with an outline project and these were assessed to work out the more feasible ones. A shortlist is then given time (and £) to work up the full bid.
Some add their own considerable financial clout to the marketing - Eden was employing people to hand out flyers in Bath last weekend apparently, as well as through press.
I'm not overwehlemed by the process, but it does mean that there's not tick boxes on the back of tickets (which is clearly a bug bear of mine).