Although I take a number of two-person trips, I also go solo. The same goes for my usual hiking partners. As a result, we all own solo tents, and tend to carry a full solo load (each has his/her own filter, stove, first aid kit, tent, and food.) This is the simplest approach for us because it's the most versatile. Yes, we could share gear to lighten our total load, but it's just easier not to.

However, if you're just starting out, and neither of you has any gear, you may want to consider sharing: you buy the tent, he buys the stove and pots, etc. It will result in a lighter load than if each of you carry a solo load. It will also mean that you can't just take off on your own if you want; you'll have to coordinate with you partner to borrow the rest of the gear - and you will be traveling a little heavier if you go alone. But, in total, the weight differential need not be more than a pound or so if you select gear carefully. (There will also be a cost savings, since you'll only be buying one stove, one tent, etc.; and 2-person tents usually cost less than 2 solo tents.)

If you start by sharing gear, then decide later that you want to go solo, it's not a big deal to add the missing pieces. And, if you're the one with the 2-person tent, the weight penalty isn't prohibitive; take the 2-person tent for a while and enjoy the palatial living!

Enjoy your hikes.