Like the bishop, the priest has a particular role in the process of initiation:
13. Priests, in addition to their usual ministry for any celebration of baptism, confirmation, and the eucharist (see Christian Initiation, General Introduction, 13-15) , have the responsibility of attending to the pastoral and personal care of the catechumens (see Presbyterorum Ordinis 6), especially those who seem hesitant and discouraged. With the help of deacons and catechists, they are to provide instruction for the catechumens; they are also to approve the choice of godparents and willingly listen to and help them; they are to be diligent in the correct celebration and adaptation of the rites throughout the entire couse of Christian initiation (see no. 35).
How do you read this? This is what I see: going the extra mile beyond the ordinary pastoral care provided to parishioners. If the parish has “troubled” persons, it is to be a special care of the pastor to address those troubles. Catechesis is a shared ministry with deacons and lay people–that’s how it usually falls in a parish. How often should a pastor or other priest be on the line to teach catechumens? Probably somewhere between every week and a token appearance. Once a month?
The part about listening to and helping godparents is easy to get lost. That’s not a usual detail for pastors working in RCIA, but it does relieve the lay staff of the exclusive role of forming sponsors. My sense on godparents is that a particular formation should be part of their role, and that the pastor should take a part in that.
Note that correctness extends to adaptation of rites. A pastor really needs to be on the discussion and discernment end of that, especially if he is presiding. Some priests, sadly, just accept whatever is handed to them.