This is referred to as going strong side.
Strong side officiating differs in some areas, and personally The Federation messed it up for a couple of years...

If you look at all the diagrams.. Lead's primary used to be below FT Line (with exception of outside the 3 point arc).. Then Fed added that goofy thing about opposite lane last year or year before where lead did not have the ball opposite lane. My opinion and I think the changes this year validate this..They did that because too many leads would not go strong side or were making long distance calls across the paint. They did not want us to stop going stong side, nor did the Diagrams change. The one still showed that when lead went strongside that they did pick up ball when it was below FT Line.

Heres my opinion for what it is worth...If the ball is in a spread offense, with lots of skip passes you are probably not going strong side much or you will be a yo-yo.. Once the ball settles below FT lane line, (or there are a couple of big guys you need to go pick up to stop banging) Go after it and pick it up. My rule of thumb is... If I can say to my self "Should I go or Should I stay?) and the ball is still there I go...

This is also where you watch teams and pay attention to how they run their plays. If they isolate you may go across and pick up earlier--

Your goal is the best officiated game. Go across and ref the ball and dont be afraid to pick it up in front of you, we do it when we are opposite the Trail, why cant we do that when we are on the same side...? You just need to make sure that as you do that you have good off ball coverage by Trail... I also think that with Lead strong side, Trail off ball Trail has a better look at he cuts through the middle to where the ball may end up and can pick up the defense better and determine the blocks etc going down the middle or other sideof paint... Trail will always have perimeter and weakside rebounding no matter what side of the floor lead is on (weakside=opposite of lead) They always have the best view