I don't know anyone who has done this and found it to be successful (except international students with funding looking for PhDs) but I do know people that have got postdocs from their supervisors contacting their 'friends' and asking them if they want a postdoc. They would then just go through normal employee enrolment.
My supervisor told me to contact random PIs directly, but like you, I wondered how to do it and how effective it would be so I didn't.