I stayed in private accommodations (small hotels/hostals/posadas) and did not feel I missed out on too much. I invariably ran into someone I knew from previous days and walked and talked with them to wherever our end stage was for that day. I ate some meals alone and some with other peregrinos. Some evenings I sat in my room and relaxed, some evenings I wandered around looking at the town.
I did spend a couple of nights in albergues and they are mostly not for me. I did enjoy the atmosphere for communal meals but was unable to sleep in a room with 10+ other people.
As you are travelling with one other person, the cost of sharing a room will be very close to what you would both pay in an albergue, although some private rooms will have one double bed and some will have two singles. I would try albergues for the first couple of nights and if you don't like it then just book private rooms.
Many of the Camino apps will list albergues and private accommodations, booking.com was my go to for rooms.
I think the pro/con list for albergues looks like this: Pro- meet more people Con - meet more people