In some cases the well is completed by drilling guys, mostly when the downhole configuration is not too complicated and circumstances are such(eg. no service rig available etc). I most of the cases though, completion is done by service rig (it is cheaper, completion sups are more knowlegeble usw.)
Later on, on the life of the well when the well needs to be serviced, there are the same "completion" guys who go there with a service rig and do the repairs.
Now, in all of above the production guys have little to say. Their input is the specs of downhole production - the other things (holesize, casing, tubing, WH are designed and executed by drilling/completion guys. Shell calles them well engineering group, some other call them operational group, but they are not part of production. In small companies there are experienced production/exploitation guys with exposure to operations who contract the work to well engineering firms.

(sorry for the late reply as I was out in the field last week)