My 2 cents. If you are new to it or unsure of what you want to haul renting makes sense until you can figure out what you want to do. For example you rent a van and then decide you would rather have a flatbed or reefer or whatever. Also the market for whatever you are hauling factors in. Are you sure there is a good market for that type of trailer in the area you want to work? With renting you are not locked in to a purchase. It might also be a good way to get your foot in the door starting out if you don't have the cash to purchase one.
If you know what you want to haul and you are sure the work is there purchasing will be cheaper over the long run. The people that rent trailers out to people are a business. They buy trailers, pay their staff, advertising, office space, etc and still charge enough rent to make a profit. You don't want to rent a trailer for years on end you will pay more rent than the trailer costs.
I don't run my truck all the time. I bought it and my 53' step primarily to haul my own equipment around but I have authority and do some hauling for hire. It also sits a lot of the time and I don't care because it is paid for. I would not want to be paying rent on a trailer that wasn't being used.