We never did put my first horse's papers in our name, he was 15, I was never going do to high level shows with him, or breed shows with him, he was a gelding, so breeding was obviously not an option, and when we ended up selling him he was just being sold as a trail horse, so there really wasn't an incentive to pay the fee to change his papers to our name. That being said, the first thing I did when I got my Thoroughbred, was change her registration, because I did want to show, and after seeing what a couple of people had to go through when their horses were stolen, I realized the value of having the papers in my name regardless of whether or not I was going to show in any type of show where registration mattered.
We bought an Arabian a few years ago, and she came with all the registration paperwork filled out, but it hadn't been sent in yet. She was a purebred mare with amazing lines, and definitely something worth breeding someday, so I wanted her registered, and since I got her from the breeder I figured no big deal, I'll just send in the paperwork, and the fees (which were doubled by this point as she was now 4 years old), and I'd be done. Unfortunately I ended up finding out that the bill of sale didn't have the right wording, and by then the breeder was having problems with the rest of her horses, and I wasn't about to try and get her to rewrite the bill of sale for a horse she didn't even remember. The person I sold the mare to fortunately has worked extensively with Arabians, and with the breed registry and was able to get her registered finally, but it was a massive headache that would have not existed had her papers been sent in when they should have been, and been kept up to date. Because she wasn't registered yet, the bill of sale needed to state the year she was foaled, her color, and her sire and dam, but the breeder had instead just put the name she wanted the horse to be registered as, and the date of foaling which wasn't enough. So anyways, long story short, definitely keep the papers up to date regardless of whether or not you decide to pay membership dues for yourself.