How it is handled will vary from situation to situation. Do whatever works best for you.

I happen to be a renter (no longer desire to maintain the upkeep on a house myself) but I am also a gardener, and a professional one at that. I have always maintained the gardens at the properties I rent with the exception of lawn care. In fact, as a landscape designer, the properties I rent generally have better gardens when I leave than they did when I first arrived!! But it is all done with the permission of the homeowners/landlords (who in my case have always been more than glad to provide).

Hiring or doing it yourself depends on your bottom line. Do you have the time and energy to do it, and save yourself some money every month, or do you prefer that someone else does it, because maybe you're managing so many other homes or properties, or maybe your family has a strong priority on evenings and weekends, etc.

After high school my buddy and I mowed our own lawn that his grandpa rented out to us, Otherwise I have only lived in apartments till now. My neighbors, however, which I'm pretty sure is a rental. They have very basic low level garden and they hire a company to mow and trim their yard. they have Iowa natives, day lilies and hostas around a couple trees in back, and woodchips with drought tolerant plants in the front of the house. With any yard, especially a rental I would do more with less.

I know there is a land lord association site that should have info if I remember it right. Otherwise you can check on the price of what it would cost to have a company to do it vs a tenant (making sure they do it) and knock a small bit off the rent.

Only let the tenant do it if they know what they're doing! You could be sorry. Likewise, some "landscape" guys just prune everything into meatball shapes and weed whack your good plants. Choose carefully by local recommendation.