I've been in both situtations, had lessons/worked for rides for seven years and now I finally have my own horse. :) I don't really think the "dirty work" is much work at all if you only have one horse. Coming from a person who usually cleans fifteen stalls, one is no biggie. ;)
Not owning a horse:
Pros - Gaining experiance from riding many different horses and not developing bad riding habits because each horse differs. The ability to move on to a "harder" horse as you advance in riding.
Less responsibility/can go on vacations and not have to pay for someone to look after your horse.
Less money spent = more money to show and etc.
Cons - Less bonding time.
Having to "share" the horse with everyone else that rides him/her.
Not being able to ride all the time.
Owning a horse:
Pros - A real bond with your horse.
You can ride whenever you want and no need to share your pony. xD
Cons - More responsibility(this can be a pro for some people)
Lack of money. :P And time, as you have more to do when you're the sole caretaker of the horse.
Developing bad habits by riding only one horse, or having either one of you holding back the other.
Honestly, I really miss riding other/many horses. Thankfully, my BO is the bestest and let's me ride her horses sometimes, but usually I don't have the time to.