There is no "right or wrong."
I drill my own stuff, and here's why (and I'm sorry if this is long):
I have 2 shops that are "close." One is about 15 miles away. The other is about 30 miles away. I think there may even be another about 35 miles away. Neither of those are in a center - they are standalone shops.
There used to be 2 more that were about 15 miles away - each in a center until AMF decided against having a pro shop, but that's a different story.
So, those are my local options. Neither shop can watch me bowl. The closer shop did all of my drilling for about 2 years - probably about 30 balls. In that time I bought 3 balls from the shop - 1 new and 2 used. All the rest were either new balls from ebay or online shops, or used balls from here.
The new ball had to be ordered because he didn't carry that brand (AMF). At the time he only carried Ebonite, Brunswick, Storm, Hammer and Columbia300. I also had him order a ball for the wife. The price for mine was about the same as ordering online, maybe a little higher - $189 $130 online and another $40-50 for the drilling. The one for the wife was higher through the shop - $150 - was about $100 online, but she throws conventional and it's only $20 for a conventional drilling. His drilling prices are now higher.
I ended up having an issue with that ball. I just couldn't get it to work. I talked to him several times and got these responses - "I drilled it stacked. That thing should hook like crazy." "I told you I didn't know nothing about that ball when I ordered it. It should work fine, maybe you're throwing it wrong."
I ended up taking the ball to a different center (AMF) that had a pro shop. I asked the shop if they could look at the ball and see if there was anything wrong. They watched me bowl and agreed - the ball looked like it was trying to hook, but never turned. They weighed the ball and found it had 1 full ounce of side weight. I guess I don't have enough revs to make that work? Anyways, they took some more weight out (already had an x-hole) and that fixed the problem.
That was just one small reason, but there are others. Every time I took in a ball for drilling (or plug and redrill) they would give me a hard time. Good natured joking or not, I've never seen a business tell a customer to stop coming in and spending money! They didn't stock many balls, and if they did, rarely did they have one in my weight (14). So, I'd have an hour into driving there, ordering (or dropping off), and then going home. Picking up the ball was at least another hour. The ball would never be ready - you'd show up, he'd pull the measurements off of a ball you brought (he didn't keep spec sheets), then you'd have to wait for it to be drilled and the thumb worked out (usually much bigger than needed....and I hate using tape!).
One day, just for fun, I took 6 balls into the shop and asked him to measure the spans. He looked at all of them, measured them up and said "they're all different." I looked at him and said "that's funny, they were all drilled here."
I did go to the other shop that is 30 miles away. I had them remeasure me - span and pitches were just about the same as the other shop (depending on which ball you used!). Also had a new ball drilled up, and had them plug and redrill another ball for me. I was happy with the results, and decided that would be my new shop. It just so happened that I ended up winning an auction for a drill press, bits and jig. A few more auctions later and I was able to get enough tools and pieces of equipment to plug, weigh, and drill for myself.
I only drill for myself. I don't pretend to know everything, and I'm sure not perfect at it. The only thing now is that if the span is off or something gets screwed up, I didn't pay somebody else $60 to do it wrong!
I'm a firm believer in supporting the local shops. If you're lucky enough to have a good one nearby that is great. My local shop didn't carry a large selection. He didn't care if you brought in an outside ball - he was happy to make the labor charge for a blank drill and not have to do any ordering or stocking.