Personally, I would not share information about whether you have other interviews scheduled. It's none of their business and does not provide you with much advantage. (They don't tell you who else they are interviewing….) If you are explicitly asked about other interviews I would be truthful but non-specific such as, "Yes, I have two other interviews scheduled for later this month. However, I am very excited about this position for XYZ reasons."
If you have an offer in hand, that is a different scenario. I would only disclose that information at the end of the interview with dept chair or search committee. Again, I would not share much but say you have a competitive offer from another institution. I did say, "I would really like the opportunity to consider this position. The timing for my decision for the offer is this date, can you let me know where I stand by then?" Generally, places were receptive to that and worked with me on the timing.
Just remember, your objective throughout this entire process is to get a job and have them excited about hiring you. Telling them about other interviews isn't going to help with that objective.