I personally love the bounty hunter missions, makes me feel like a right bad-ass haha. I always like to be sneaky and then let rip, snatching the bounty alive.
I think its spot on what Debbo89 said, it doesn't make a whole lot of sense for Arthur to be doing this from the law perspective,but there may be missions
very similar to bounty hunting given by Dutch or another well respected member of the gang to hunt down people who have wronged the group, or those that
are standing in the way of Dutch's plans for the group. That'd be really fun and surely build trust with Dutch ad the group. It's gunna be interesting to see if you have a 'choice' in these sorts of things or whether they'll be part of the main story. I think they'd probably be 'side' missions though, that may affect the story telling.
for instance, dutch could give you a side mission to go get intel on a bank the groups planning to rob, before the actual heist. It'd be cool to have the option to walk in all guns blazing and not know details, or have the choice of going on an extra mission to find out info before the hesit. This could also weild benefits and extra goodies as youve put more work into the job.