There are two possible scenarios:
- The faster accelerating person is in front.
- The slower accelerating person in in front.
In case 2 there is no tension (the rope is not taut).
In case 1, if the rooe rigid it will break. Otherwise the different accelerations are not possible. Alternatively, if the rope is elastic, then elastic tension (spring force) will grow since the two people will get separated more and more, elongating the rope.