It varies.

Some of the Tank Classes make it very difficult to retain aggro.

Sometimes the Healer seems to pull aggro - at which point of course, everyone should attack that mob as it is essential to protect the Healer.

Sometimes one of the DPS just rushes on too soon and gets aggro. A good Tank will work to take that back again.

Of course, a good Tank will ask the DPS not to do that and a good DPS will try not to attack anything the Tank has not already aggroed.