My question is how does one do bounty on the managers website? Clear me, cause I don't understand what you mean by that. No bounty runs on their own website, if they run it on their website they are not going to get audience. So they either make use of forum (bitcointalk) or they make use of social media or they can even make use of the both of them. So do you mean social media?? If that's what you are referring to as the managers website then I don't see anything wrong with that. There are even jobs where you tweet for Bitcoin, so it's just like that.
I think you don't get the grasp here. Bounty managers are offering a sort of portal in their website wherein that's the place you will apply instead here in the forum.
But definitely, the task is to be done here in the community, specially if the bounty you joined is signature campaigns. And managers have the tools to count everything, reflect it in their portal so it eases their job and bounty hunters will have to go and check their stakes on that said portal/websites.