The answer is the same as it is with people. It depends on the dog.
I grew up doing the same thing you did. We'd simply refill the bowl when it got empty. It was fine with our first lab. She was leaning more toward heavy than skinny, but was a decent weight. My second lab turned into a butter ball. I had a knock down drag out with my parents at the time, because she'd been hit by a car and as she got older, the arthritis took a toll. It was made much worse by the extra weight. She should have been about 73 lbs (33 kg) according to the vet, but she was more like 93 lbs (42 kg). My parents still wanted to feed her all kinds of table scraps.
I'd come home from college for the summer and put her on a vet recommended feeding schedule and walk her a mile or so several times a week, as well as regular weekend trips to the river. I'd get her down to 75 lbs (34 kg) or so and she'd be running around and playing like her legs didn't bother her. Then I'd go to school and they'd fatten her back up the 90s lbs. She'd limp around and not do much. I finally convinced them to quite and she did much better.
Now I have a Jack Russell and a Keeshound I inheirited. I feed them a measured amount twice a day, but they often leave some in the bowl till the next feeding. I could leave it in the bowl for them 24/7 and they'd eat the same amount.
Basically, you need to monitor your dog. If he's getting fat, then you either need to exercise him more, cut back on his feed, or both. Either way, I'd recommend doing it gradually. Consult your vet to see what a proper amount of food is for a healthy dog of his size and breed. Then buy a higher quality dog food. You don't have to break the bank, but you'll see a noticeable difference in coat condition and other areas if you feed a higher quality.