I lived in a rented duplex house (side by side apartments) where the previous other tenant and I took turns mowing the lawn using his lawn mower. It was never an issue. Sometimes I mowed the entire lawn and sometimes he did. Sometimes we did it together. When he moved he took his lawnmower with him.
When the new tenant moved in she announced to me that she had had a lawnmower and if I wanted to use it I would have to pay her $10 each time I used it, but if I mowed “her half” of the lawn I could use her mower for free. I found that to be not necessarily rude or mean, but very un-neighborly. I grew up in a neighborhood where lawnmowers and other tools were shared. The sound of someone taking our extension ladder or lawnmower, etc. on a Saturday morning didn’t give us a moment’s start.
So I purchased my own lawn mower and proceeded to mow “my” side of the lawn and “her side” for about a month and a half. I made sure she was home when I was doing it. Eventually she apparently felt guilty and approached me and offered to alternate the lawn mowing duties.