I have access to the Pets private beta. Someone has recently added a bounty to their question.
Since this is a private beta with at least a week and a half till it hits public beta, the visibility of the question to potential experts outside the beta is quite limited, in my opinion reducing the effectiveness or even need of the bounty during this phase of the site life-cycle.
Is it necessary to have the bounty system enabled during a site's private beta phase?
If people want to spend their rep in that way, knowing the limited audience, why stop them? Every per-site or per-type-of-site customization has a cost in implementation and maintenance (and possible interactions with other features), so this should only be done when there's a strong argument. I don't think "that wouldn't be an effective choice" is a strong-enough argument.
Further, I can think of a reason to offer a bounty during a private beta: to reward an exemplary answer (one of the standard bounty reasons). While most bounties are there to attract new answers, don't forget that there's another use case. And, particularly on a new and growing site, someone with a lot of rep might choose to give some of it away to worthy answers in order to help other users gain privileges.
I'll answer generally, as this isn't specific to pets, but more to how private betas work.
The Area 51 process is supposed to guarantee that a small, but sufficient core of users will be on the site and actively building it during the private beta. We'd be a little worried if a bounty went unnoticed during this time, because it might suggest that not many real experts made it to the site during the most critical time in its history. That of course depends on the question, some questions require experts with extremely rare knowledge to answer correctly, or the question might just be difficult to comprehend.
While not yet open to the public, there should be (at least in theory) approximately 200 people active on the beta during the first week, so there should not be a lack of visibility. While the question is initially only visible to those that committed or were invited to the proposal, it should not be a trivial audience.
Also remember that bounties can start in the private beta but not end until the public beta, depending on when they were placed. It's not often used so early, but the bounty system should work as designed during the private beta phase - especially on a site like pets that saw a very healthy turnout.
There's also, as Monica noted the utility of giving an additional reward to an exemplary answer. It's important that votes and rep are flowing healthily during this phase, as folks need privileges in the next one :)
tl;dr; - There's no reason why it should be disabled, and having it enabled is actively helpful.
One of the purposes of beta releases is to exercise as many features of the final release as possible under realistic conditions. That's a good enough reason to enable the bounty system feature. If the bounty system on this site had a bug or a usability issue, why not give yourself a chance to discover and address it now while the audience is smaller and more forgiving?
...why bother to disable it? It doesn't hurt anything by being enabled and it is an extra check to add to the system.You aren't "enabling" bounties on non private beta sites. You have to actively disable it on private betas since it is part of the system. So why bother with the extra overhead (although probably minimal) to now check to see the status of site before permitting bounties.