Do student hosts play a part in the application process? IE, does their opinion matter, do they report back to the school, etc.
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Almost always, the answer is no. Your student host won't be the same student that interviews you, and doesn't give official feedback to the committee. However, I'm sure if there are rare cases where the host really likes you or really doesn't, and mentions something to this affect to people he knows on the admissions committee. Therefore, it's always a good idea to stay on good terms with everyone you meet when you visit a school.
99% of the time - no. They are only being hospitable. However, the student member of the adcom is usually well-connected and very sociable, so if you make quite a bad impression, word will get around. But then, the student member of the adcom has very little say in the final decision to accept or reject...
This is definitely not true for all schools. At many schools (eg, Hopkins, Pitt) the student interview counts just as much as the faculty interview.