Games and movies and such make it seem like bringing in wanted criminals dead or alive was a normal part of the Old West. Was it really that common for independent freelance bounty hunters not associated with any organized law enforcement to just run around looking for wanted posters?
So a lot of time the people collecting the bounties were deputized. Take U.S. Marshall Bass Reeves, for instance. He worked out of Fort Smith, Arkansas and routinely ventured into Indian Territory (present day Oklahoma) in search of outlaws. Those he brought back he was able to collect bounties on.
I don't know the actual numbers, but I'm not aware of any of the old west figures who were full time civilian bounty hunters. I'd say that most were working in law enforcement is some capacity.