I wrote one to my school this year. I was waiting and waiting and waitinggg for the first call for an interview (*keep in mind, though, I didn't submit AACOMAS til Sept. 15...I was verrry late in the game). Anyway, I decided to take some action and start calling schools. I contacted one of the particular schools which does not give out secondaries until you have been invited for an interview (WVSOM). I decided it would be wise to write a "letter of interest" expressing just that, my interest...and my desire to attend and my respect/love for the school and surrounding community. I did this, mainly, because I knew that without the opportunity to fill out a secondary, I had no "extra pull" to help clench that interview--and no means of expressing the special connections I have to the state, as well, which I felt was important to convey since I was an OOS applicant. After sending my letter, I ended up getting an email 3 days later about scheduling an interview.
I can see this letter of interest being a wise step, as well, if a school is taking a long/longer time than they had intially said to make a decision on your acceptance POST-interview. Also, maybe if there is something that you did not have the opportunity to address in a respective school's secondary, you could state in a supplemental letter (ie, like me, as an OOS applicant, you have a special connection to their state/community). Once you've submitted that school's secondary, I'd say to wait to write the letter until some time has passed with no word from the school. Writing that letter could help you to get a "little gold star" on your app for the committee to double check/re-review it. Time will go by where applicants are kind of just placed in a "deferred" or "set aside for now" pile; these are the kids that aren't immediately granted the interview. This was me. I was in a "decision deferred" pile from November - March at WVSOM. By sending that letter, you may just get yourself out of that pile if you hit home with your words, with the right person reviewing it.
Be sure to express your respect for and connection to the school...and, esp for some schools, demonstrating your interest in becoming a part of the surrounding community and their respective medical world is a great way to convey a sincere interest. Of course, you should mean it 😉
Hope that helps! Good luck!