A lot of factors play into how I regulate substitutions.
First is what do the tournament/gaming rules tell me. Do they specify that the player must be all the way off before the other player can come on? If so, than I have no choice how I run subs. If the rules only say that the sub may only enter with the referees permission, than I have leeway over how I control the subs.
Second is the length of game. If the games are full length, than I am strict about substitutions and go by the LOTG. Players must be fully off the field before the substitute can enter at midfield with no deviation. If the game has a short half such at 25 minutes with no add time for U14, than I am going to bring the sub on quickly. I do not care where the player exits the field (I prefer they do not exit in front of the other teams bench), but they must enter close to the midfield line.
Third is the age of the players. If the players are high school age, than I am strict about substitutions and go by the LOTG. Those few extra seconds during substitutions allow me to take care of game management issues such as to talk with players especially a player who is on the edge of getting a card, or to check on a player that may have got a minor injury. It also gives me a short break from the running since my 50 year old legs get tired after running around with 16-18 year olds after 2-3 90 minute games.
Fourth is the experience level of my AR's. With newer AR's I will usually be stricter about subs so the AR has a chance to learn the proper signaling techniques and controlling the subs. More experienced AR's know or should know how to control the subs.
Fifth, and a pet peeve of mine, is that the substitute MUST be up at the midfield line and have checked in with either the AR or 4th official. If the coach is yelling for a sub and there is no player standing at midfield and no injured player on the field, than I will not allow for a sub. This usually only has to happen once, maybe twice, before the coaches figure it out. I always instruct my AR's to brief the coaches before the game on the substitution procedures and will not start the game until I know they have talked with the coaches. That way the coaches cannot say they didn't know. If I am going to be strict on the subs, than I will also tell the players during the pre-game player check-in.
*** This is something that I always do and wish more referees did. I wait until the sub has come onto the field and gets into position before restarting play. I have seen far too many referees start play while a sub is still running 60 yards across the field to get into position. Not waiting for the sub to get into position is unsporting and provides an unfair advantage to the other team.