The most common mistake that I see people do with their puppy is start the off leash portion of their training too early. Puppies make owners overly confident that they will stay this way, by the natural instinct of a puppy to stay close to their leaders. However, early lessons that they can get away with their puppy speed, will unfortunately carry over as they grow into adulthood and make their natural curiosity and wander lust worse than it needs to be.
It also starts to inhibit any problem behaviors related to prey drive. In puppy hood you can begin to teach the skills not to chase small furry animals, large furry animals, and cars. All of these things have the potential to be deadly to your puppy and future dog.
At the point that dogs are puppies is a great time to start the checking in process that you are going to desire before allowing your dog to go loose. I do this with puppies or adult rescue dogs with almost no training by doing the following:
1. Inside, while not supervised (definition being complete concentration on the puppy or dog), they are either tethered to me OR in their crate. When I am stationary, I provide a dog bed and toys that they can play with. This is not the sit on the dog exercise, which limits the dog to an area just for quiet. Be aware of where their teeth are in relation to the leather lead, as puppies and adult dogs are fully able to chew through a leash with their back molars very quickly. Otherwise I let them hang out and play around me. If they are tethered to you, you are likely to become aware of when they are in distress of needing to pee or poop. You will also be alerted to their movement immediately.
****NOTE: If I have a puppy that I suspect will worrying the leather lead and I doubt my ability to be aware of it, it is acceptable to use a rope for an indoor ONLY lead or old belt for an indoor ONLY lead for this so you can both relax, and you don't' accidentally loose something valuable. Most puppies are not so stubborn in doing this, but I have had a few small terriers that managed to get their way (and one Thai Ridgeback). So while I was teaching them not to do this, I was using less valuable leads for the inside puppy training. Some puppies are more determined than others.
2. Inside while supervised (definition being complete concentration on dog or puppy), I do allow them to play off lead and most likely with me or a family member. If they are content playing with toys alone, I allow them to do that. Do keep an eye on them though, because they can become bored with that toy and wander off to do something bad. I have a bunch of play sessions that I schedule into my day to allow them this sort of freedom. With a puppy these can be limited to 15 to 30 minutes, and is a good bonding time for the puppy and owner. IF there is a possibility that they are going to be naughty while supervised or have an accident, it can be very handy to have them dragging the leash inside. It is sometimes much easier to step on and grab the leash, than to need to pursue a puppy to catch their collar.
3. Outside, I would only allow a puppy to be completely off leash in an enclosed area unless in a rural setting where you are 100% confident that puppy is staying with. Otherwise, they are most likely dragging a long line. This way you can gently and efficiently stop any dashing off by stepping on the line or grabbing it. This starts the process of the puppy at least starting to ask or check in with you first before you release him/her. You do need to watch for the long line tangling around other dogs and themselves. Be sure that you don't do this near high traffic areas, just in case. It's not in most puppies natures to become serious flight risks at this point, so catching a 20 foot leash should be relatively simple.
4. Bathroom times, I will bring them along on a long line as well (sure to flick the line out of the way) unless I have an enclosed area to let them out in. In some cases, if the owner needs to have their dog eliminate on a six foot leash (they live in the city or they want it to happen on a walk in an area that has a leash law), it is prudent to start that training now. If dogs are given room to go to the bathroom, which I normally have no problem with, it becomes difficult later on to teach them to go on a six foot. I don't think that is your situation, but it is something to think about, if this does apply to you for some reason. (for instance when you take him to where you work, you might find that you have this need)
Although it sounds like more work now than one would want, it saves you a lot of aggravation later on. So many owners come here with dogs that have never been on a leash. When you have large areas to let your dogs off at home, it's easy to forget that you may come across an unexpected distraction when out. Also as you discovered, that distraction can live right across the street from you. Now is an excellent time to ensure your dog has more freedom in the future when it counts, and less while they are very young puppies learning the ropes.
It's also what starts this sort of calm and confident personality in your dog (in balance with the crate training providing a bit of independence early on). That demeanor in your future dog helps people have more confidence in your dog and your ability to handle him in public. Your future dog will have lost none of his ability to have fun and rip it up as needed either.
Remember, you can't do off leash what you can't do on leash first. This is really important to remember in order to keep you from unnecessary aggravation later on.