At this time of year, some of my colleagues and I like to show each other our favorite "rejection letters" from the recent crop of grad applicants -- that is, the letters that we get from prospective graduate students who decide to accept an offer other than the one from our own department.
I am sure that these letters are difficult for students to write, and it already shows a degree of thoughtfulness to send one. A personalized rejection-of-offer is of course not required; students can just click 'decline' on a webpage and be done with it.
Even so, it's nice that some students send a note of some sort. Some students send a lot of e-mail to potential advisors during the application process, request individual meetings at conferences, come for visits, and basically need a lot of time and care from faculty as they (the students) collect information to make their big decisions. It's polite to thank someone for their time, whether or not you decide to work with them.
The mutual-sharing of entertaining rejection letters is therefore not a mocking of sincere students, but just a weird professorial habit of laughing about some of the stranger aspects of advising (or not advising) graduate students. In particular, some of these letters are interesting for the contortions the students go through in an attempt to let us down easily as they explain that we will, unfortunately, not have the opportunity to work with them -- at least.. not directly. Some students comfort us with the possibility that they will stay in touch, we will see them at meetings, and we will get to see how it all turns out for them in the end.
I have written about this before..
In short, my preference is for a brief and sincere thank you. It's also nice if there's a mention of where the student has decided to go, but there is no need to explain why that was the decision.
In particular, there is no need to reassure a professor that the applicant really does respect them and their work. We do not need to be told that we do interesting research, but.. These 'I actually think you do really good work' e-mails from students annoy at least one of my colleagues, but I think most of us recognize them as classic examples of an academic- letter genre and appreciate the thought, if not the awkward language.
8 years ago