Let's stay focused on calling priorities here. What we do not want are two officials making calls on the same play at the same time. This is why this is a staple of the pre-game conference. To steal the NCAA mantra 'call in your zone, see in all zones'.
As Mike pointed out, it is the plate umpire's responsibility 'near' the plate. Defining 'near' will be determined at the pre-game by the officials (mine happens to be more that two to three steps). The 3-foot lane is often used as a guide because you have a physical representation to look to. However, it was also mentioned that this is seldom available. When I see a play made in the 'grey' area (when the play is moving away from me) or beyond my calling area, I immediately look to my partner. If s/he does not make a call, now I step in and make the call. In these 'grey' cases, I always hesitate and defer to my partner but am ready to:
1 - Make the call if my base umpire does not
2 - Make the call when my base umpire comes to me with a live ball appeal (this looks great, by the way)
3 - Have a discussion with my partner after a coach has excersized her/his appeal.
Ideally, the call is made correctly when the play happens. But inserting yourself into a call, even just to assist, can have consequences. For example, if the tag was missed, I don't jump on a missed tag call, because my partner may make an out call
. Or, as you mentioned Blu, call 'TAG' and have my base umpire calling no tag. Unlikely, yes, but not impossible.
An aside about the live ball appeal -
I like the live ball appeal on (sweep) tags on the line and pulled foot at first. If the base umpire has an out, but needs to certain of the last piece of the puzzle, and the plate umpire has the angle and position to see it, this works great, and as I mentioned, looks fabulous. There may be no better show of good teamwork than when this mechanic is properly applied.