I'm curious. Can laypeople in the pews privately offer up a Mass for their own specific intentions? I know priests can offer up the mass themselves, and I'm wondering if offering up Mass is reserved to the ordained.
Yes during when the priest elevates the host and chalice you could pray something like “Oh, Lord in union with the sacrifice of this mass I offer up …”
Yes we can offer a Mass for our intentions. You can tell God your intentions even before the Mass begins, but I recommend this to be done during the "offertory".