Allowing a student to sit in during class

If you want to find out what exactly is within your legal power as an instructor and what is not, check your university regulations. They do vary from place to place. The general rule is that if there is no official policy in the handbook, the choice is yours, but you may also ask the chairman to be on the safe side (this is a good general strategy, by the way: while in doubt and seriously concerned about any legal stuff, just pass the issue up the administrative ladder). As to the rest, I usually allow people to sit in whenever they request it under two conditions

1) The classroom sitting capacity allows it

2) The student behaves in a non-disruptive way

So, unless you have a clear reason to say "No", I would say "Conditional yes" and tell the student any conditions you care about right away with the understanding that if any of them get violated, the sitting-in gets terminated.

Assuming that there are no university or departmental policies preventing it, you should consider the reason why the student would like to sit in on the class. I have had many cases where I have allowed non-enrolled students to attend my class. For instance, I have had several students sit-in on my doctoral courses, because they were trying to decide if they should apply to the program or not (one of these cases, the director asked me to allow the person to sit in, if I was okay with it). Some students have indicated that they have friends or colleagues with specific interest in the course topic and have asked if the student can join. I have found that acceptable, too. I have also had one student sit in on an undergraduate course, because he was deciding which school to attend and wanted to and was interested in the major that I taught in.

One case where I did not allow it was when I had a highly disruptive student who disapproved of course content (including content that I was required to teach in the course) and periodically became combative during class meetings (my director was aware of this ongoing situation). I was highly concerned that this visitor would further disrupt the course lecture or discussions and I did not allow it. So, consider how the visiting student could or may inhibit your teaching goals.

If the student is interested in attending an extended number of classes, I may also frown upon that. In my cases, they were one-time occurrences.