It only alters outputs when the last layer of the neural network give significant weights to multiple tokens, and it would anyway have picked a random answer.
Instead it picks a non-random one, but non-random in such a way that you can't tell without the private key of the watermarking.
This mostly adds randomness these days for branches in syntax that make no difference, and the model has no reason to believe make a difference. Anything that matters, it is much more confident in the last layer of weights on the token to use.
Others have already said this, but the watermarking is something like "when the model flips a coin picking between two values, always choose heads". It was already flipping a coin. You're not choosing a less good result, you're just using a deterministic process when it was stochastic before.
This will have some impact on outputs, but unless you have some reason to believe that always picking tails was better than always picking heads (in which case, you should be working at one of these companies in model training!) it won't have any impact on output quality.