C172 Flight Instructor reported the LRY and LWC airports are referred to using the same name and share the same CTAF; with similar airfield layouts that resulted in confusion with pilots and ATC. Flight Instructor also reported these two airports are 47nm apart creating radio overlap.

Narrative

We had planned to conduct a flight departing from ZZZ to operate in the surrounding area to the west of ZZZ. We decided to make a landing at LRY; Lawrence Airport; for bathroom breaks and while enroute to LRY we were monitoring the CTAF frequency for LRY of 123.0 and heard multiple calls of traffic in the pattern at Lawrence. These radio calls of traffic in the pattern at 'Lawrence' indicated to us that the Lawrence airport was operating as expected. As we joined the pattern to prepare for landing we noticed there was no runway visible. The runway had been destroyed and was apparently being completely rebuilt with heavy machinery in place doing the construction. There were large yellow X's at the end of where the runway would have been. We were still hearing traffic calls in the pattern that were talking to us as we were in the pattern but we never saw any other aircraft. We soon realized that we were making radio calls for Lawrence Smith airport in Harrisonville Missouri (LRY) and the other traffic was at Lawrence Regional Airport in Lawrence Kansas (LWC). These two airports; LRY and LWC; are only 47nm apart with the same CTAF frequency (123.0) and the same name in use; Lawrence. They also have similarly aligned runways (LRY 17/35 and LWC 15/33 or 1/19) which are close enough in alignment it is easy for people to confuse or misspeak when making radio calls. We thought the traffic we were hearing was just calling out the incorrect Runway 17 instead of 19 when making their position reports in the pattern. The normal radio calls such as 'Lawrence traffic; Aircraft X on left downwind; Runway 19' is heard by traffic at both airports. Pulling up the NOTAMs on our EFB for LRY we found a single NOTAM that would affect VFR flights which was a simple runway closure that began a significant time in the past (months ago?). We did not see any NOTAMs earlier because the aerodrome was not NOTAM closed. By all indications with the radio traffic for the Lawrence airport the LRY aerodrome should have been open/operating. When hearing traffic in the pattern we thought it was a leftover NOTAM that had not been cleared as the aerodrome indicated it was open even though it was not. If the only runway available is closed for several months the NOTAM needs to indicate that the entire aerodrome is closed.I recall hearing this Lawrence airport thing being a problem several years ago with ATC for an aircraft departing from ZZZ1. I heard a VFR aircraft inform ZZZ1 Tower they wanted to do a touch and go at ZZZ1 then depart southeast to Lawrence. They were approved for the touch and go on with TURNOUT TO LAWRENCE. As the aircraft made its touch and go then climbed out with a LEFT turn (east) towards their planned destination of Lawrence (LRY) the Tower gave them an instruction of an immediate right turn and course correction toward the west to avoid traffic that was launching off ZZZ1. The Tower handed them off to Departure and Departure tried to give the aircraft a south-westerly heading. The aircraft had to correct them that their heading to their destination was in fact a south-easterly heading. A quick discussion between the aircraft and ATC got them all on the same page and going the correct direction. Apparently neither the aircraft nor the ATC Tower or departure personnel at ZZZ1 were aware that there were two airports with the same name in the area. The aircraft had done as they had planned and were instructed with the turn toward Lawrence. ATC had given what they thought was a good instruction with the turnout toward Lawrence. With neither party knowing the conflict of two airports with the same name there was little chance of catching the problem until something such as this happened. If one of these airports were to use a different frequency; I imagine it would virtually eliminate the problem of two airports with the same name being confused on the radio.