Hmm… First of all let me say I won’t be quoting anything official or talking from any experience with anything official in the matter. Whatever I say is just my reflections, pondering the question now that you ask.
The question in my view suggests a preset requirement that the prayer be deep. Or in other words it’s about
deep prayer, not just prayer per se. One would have to ask himself the question why deep prayer, not just prayer. What the “deep” word is doing there and why. In my own case, I would question myself why I wanted to prayer to be deep.
Another concern is intimacy. Deep prayer together is intimate. A dating relationship is not. It may in some cases be “normal” with friends, but dating isn’t just about friendship, if at all. Again, I would question myself why I wanted to pray with that person.
Another thing is who would lead it? Taking turns maybe? I certainly wouldn’t like to become a newly met girl’s director in prayer and I doubt I’d like to be directed by her. Traditionally, in those old family structures, perhaps the husband would lead prayer - but certainly not a boyfriend, probably not even a fiance. Taking turns maybe, as I said, but wouldn’t that be making a two person prayer circle?
My yet another concern is the compatibility of the prayer idea with the concept of non-exclusive dating. Non-exclusivity suggests potentiality rather than actuality. In other words, the person is being considered for something rather than in an actual relationship (an actual romantic relationship with multiple partners is called polyamory and I cannot reconcile it with Christian doctrine and morality no matter how chaste people would make it). A prayer for good discernment could perhaps be in order, but would one really talk about marriage with a person introduced a month ago and currently going through a fifth cafeteria visit together?
Yet another problem of prayer partnership and non-exclusive dating is the premise that all the people minus one will be dumped at some point. Getting over someone who’s a prayer partner may be quite hard. Besides, there can be some messing feelings with religion. One doesn’t want to start hearing voices and all. At any rate, you won’t be holding hands and praying with those people anymore after you choose one of them (another reason not to do non-exclusive stuff if you want my opinion, but let’s not start another debate on this).
I might be inclined to enter a church or kneel at a road chapel when passing by one but I don’t think I would start praying aloud or grab the person’s hand. It would have to be my girlfriend or intended for that. It would be tricky with a female friend, as I might be inclined to do that, while I surely wouldn’t with a male friend - this breeds the question of purpose: Why. Why. At any rate, I’d probably be saying a fast
Pater noster qui es in caelis in a steady voice and without too much intonation rather than opening my mouth in the vernacular or deeply weighing the words or making pauses. Similarly, I would prefer rosary meditations written by a third person, preferably a priest or saint, rather than making up any special ones “about us” (which I do on my own but wouldn’t together - at least not aloud). Ideally, I would be kneeling down on both knees at a short distance, in one line, looking at the Tabernacle or at least at the same point but not at each other (as an offshot, remember the Exupery quote about love being not looking at each other but looking in the same direction?
), normally not holding hands. Once again, I would be careful to avoid any leading, so as not to act as the person’s spiritual or prayer director and avoid dominating her.
I would also avoid creating illusions of ministering anything to each other or taking any oaths on an impulse or establishing ceremonies of an uncertain nature, as much as certain ceremonies and patterns might perhaps be beneficial between spouses.
Perhaps the last thing that comes to my mind is that people who pray together may appear holier than they are to each other, especially if the atmosphere is deeply spiritual.
Maybe the really last thing would be that some time ago my dad said, “pray for her, not to her.” And he didn’t mean taking her picture and praying to it, obviously.
The decision is always yours, however, and I have no authority.
Personally I think praying the rosary together would be very lovely. I think when we pray it keeps us focused on God’s plan not our own. It should help you to lead a chaste life with your love until after marriage.