Light aircraft Instructor Pilot reported an NMAC with a light twin in the pattern at C25.

Narrative

I was sitting right seat in a [light aircraft]. After multiple patterns at ALO; we advised ATC we would like to return to our home base of Waverly IA (C25) and were cleared on course. We made an inbound call on the CTAF after being handed over to the Waverly IA CTAF frequency by Waterloo (ALO) ATC; approximately 5 miles out. We also made CTAF calls on downwind; base and final for Runway 11 at Waverly. We were the only aircraft in the pattern at Waverly. The left seat pilot executed a touch and go which was uneventful. We turned crosswind at 700 ft. AGL and downwind at 1000 ft. AGL. Just as we were about to call downwind; another aircraft called downwind for Runway 11 at Waverly. We scanned frantically for the other aircraft. I wanted to execute a breakout but I was concerned we could turn right into the other aircraft. I believe we called that we were also on downwind for Runway 11 and continued looking.Suddenly the left seat pilot reported he saw the aircraft below us and slightly to our left. We executed a pattern breakout and re-entered a left downwind for Waverly. The other aircraft did not make any further calls in the pattern. We continued searching as we were on the 45 to entry when I finally spotted the other aircraft on the runway at Waverly. It wasn't until after we reported we were re-entering a left downwind that he finally called he was going to back taxi on Runway 11 at Waverly. At that point I called the [light twin aircraft] and he said he needs to give an inbound call before he enters the downwind to help de-conflict with other aircraft. He claimed the downwind call was his first chance as he was just cleared to switch frequencies by ATC moments prior.I went off frequency to query Waterloo ATC about his story. Waterloo ATC reported to me the [light twin aircraft] was on an IFR flight plan when they lost contact approximately 15 miles out. Waterloo tried contacting the [light twin aircraft] with no response. I planned to discuss the incident with the [light twin aircraft] pilot but he departed C25 later that evening before I could discuss it with him. Fortunately no one was harmed and a good lesson learned for me and my fellow pilots to constantly clear in the CTAF traffic pattern and also just how important those inbound calls are when flying into an uncontrolled airfield.