Generally the platforms track the users based on IPs, cookies, etc. Then when the conversion pixel fires, the platform will see if that person was ever shown an ad. If so, it will attribute. Different channels/platforms work a little differently but that’s generally how they work.
So you can have a scenario where a person converts but was never exposed to either ad. In that case the pixels fire but don’t attribute in the ad platform.
If someone saw an ad from FB, then converted, FB will attribute and LinkedIn won’t.
If someone saw an ad on FB, then went to LinkedIn and clicked the ad, then converted both will attribute. This scenario is what a lot of people concern themself with (especially with media mixed that have a ton of channels and spend going on) because each platform will be greedy and count the attribution for themselves. So marketers need to consider attribution models, attribution windows, goal funnel paths, etc to get a better understanding of which channel is ultimately driving the impact. Especially if you count things like view through conversions the data can get a bit hairy.