The research topic says that the world elite supports both, but it doesn't say that it's the same government officials/wealthy individuals supporting both. Presumably, the majority supports X-com openly while a shady, smaller group funds the MiB in secret, though there could always be a few people that play both sides figuring they'll just deal with whoever the winner is. Also, given the aliens attempts at infiltration, it's very possible that the MiB is working with the aliens to some extent.
As for killing people, just about every secretive government organization in real-life has done illegal and immoral things and then covered them up, justifying their actions as being for the "greater good" and there have been plenty of governments that knew this happened and were okay with it. Having a government/power elite group "silence" certain people is sadly not too unrealistic.
Based on the information given, the MiB seem to be much more akin to EXALT from X-com 2012, an organization that works with aliens for their own benefit, as opposed to the Cult of Sirius, which is worshiping the aliens and actively hoping for the aliens to win. They want to bargain and work with the aliens, not get conquered by them.