Whose line is it anyway?
 

2 person mechanics, what is the proper way to cover this?

The dribbler is near the end line midway between the nearer lane line and the side line in the T's area. She dribbles into a double team and picks up her dribble. In turning to find somebody to pass to she steps on the end line.

Should the T whistle this since he is on-ball? Or should the L be looking here too in order to pick up the OOB?

A Straight Line for a Change
 

It's L's line, though it's T's area. Seems this is might be one of those times when the L has to maintain attention on his primary, yet remain conscious of where the action is.
Eyes in the back of the head help a little, too.

The T should not be looking at the foot for out-of-bounds. He/She should be looking at the players to look for pushing, holding, etc., or for any possible held-ball situation that could happen. The L, who should be looking inside the "paint", could sneak a peak in that direction to see if the player steps OOB. Use of peripheral vision is helpful here, but I know not everyone is good with the use of this vision, but rather direct vision.

IMO, this is a gray area in two-man, double-team in that particular corner as described. If I was the T, I would not be looking for OOB, but if I happen to see it, and know 100% sure it happened, while looking at the players and I feel the L may have missed it, I'll call it most of the time.

Definitely one of the limitations of 2-person mechanics, especially if there's anything going on in L's primary. Best advice to L: rather than pinching the paint here, go wide, look across your primary with peripheral vision, and watch your line.

It's L's line - there's no way I'm calling any line violation on the endline if I am T. I've seen too many times when someone calls OOB on a line that isn't theirs and the partner ends up telling them it wasn't OOB. Sometimes things look OOB from far away because you have a terrible angle, but they are not. Respect your partner's lines. You're more likely to be wrong than right.

PCA Versus Line Responsibility, The IAABO Way ...
 

For the first twenty-five years that I officiated, the NFHS two person mechanics were simple. As the lead, you had the frontcourt endline, and your sideline, all the way into the backcourt. As the trail, you had your sideline, the division line, and the backcourt endline. I never really realized that the these boundary responsibilities didn't really align with our primary coverage areas, since both officials had to know where the ball was, and where your partner was. The toughest call was on the lead's side, in the corner formed by the lead's sideline, and the trail's division line. If the ball went out on the sideline in that corner, the lead would occasionally have to ask for help, or help would be offered, or we went to the arrow, mechanics in place to help us get this call correct almost all the time.

Over the past few years, IAABO has come with an "option". The lead can, as an "option", only be responsible for his sideline out to the foul line extended. Great coverage in a halfcourt set, but not in transition, or during a press, or press break. Also, IAABO doesn't give any alternative to this option, in other words, the manual doesn't even bother to explain the old NFHS mechanics as the original mechanic to this "option". Not very good guidelines for a rookie official, especially for plays in transition, or during a press, or press break.

Our interpreter has given us his interpretation of this new IAABO mechanic. I wish that IAABO would incorporate it into their manual. He has told us that once the ball "settles" in the frontcourt, when the ball is in that corner, especially when the ball handler may be double teamed, and the trail has moved way over to the ball side of the court, if that ball goes out of bounds on the sideline, above the foul line extended, in front of the nearby trail official, that the trial official may make the out of bounds call, and then stay there to administer the throwin, as the lead switches over to the weak side of the court on his endline.

Great, but it has almost nothing to do with the OP

What's My Line? A Mark Goodson, Bill Todman, Production ...
 

Correct, but it does relate to the thread's title: "Whose line is it anyway?"