When we speak about infection, we are either thinking about microbes, virii or prions.
Virii will only infect aliens if those aliens have something akin to our cells in their phisiology. The virii would also require the aliens to have the same receptors in their cells' membranes in order for virii to work. So this relies heavily on aliens' biology being based on DNA, or at least RNA, which might not always be the case.
Diseased prions do their nasty work by converting sane prions into a bent shape. If the aliens don't have prions, these are out of the question as well.
When you get to bacteria and protozoa, though, it's a whole other story. Unlike prions they are not zombie protheins, and unlike virii they don't get into your body to steal your ribossomes. They hang inside you because you are a good environment for them to live and reproduce.
In fact, most bacteria within us don't cause any harm, and some are even necessary for our good health.
Now, imagine that you have cavities. Those are caused by bacteria eating through your teeth enamel. For us humans, specially after the 20th century, these bacteria are just a nuisance at worst.
But imagine that the alien you bit has a carapace made of the same material as your tooth enamel (but you happened to bite an unprotected, soft body part). If that alien doesn't get treatment, they may end up with serious dermal injuries over the course of a few weeks.
There are many species of bacteria that live in our surroundings, but don't thrive within our bodies - mostly because they can't handle the competition with the ones we do carry inside. If aliens came in contact with humans, and the aliens' bodies turn out to be a fostering environment for those bacteria, then the aliens will suffer infections very often. If the aliens have any equivalent to our bacteria in their world, then we can add "and vice-versa" to it: what is harmless for them might be dangerous for us.