It sounds like you are asking for some good methods to level the playing field between casual players who have modestly differing collections.
It also sounds like it is just the two of you.
The key is to have card pool and build rules that you have both agreed to ahead of time.
You need not stick to the tournament designations of Standard or Modern for your card pool rules, although those are places to start. You could use "nothing before Magic Origins". Another idea is something like "Commons and uncommons must be Modern legal, but Rares must be Standard legal." If neither of you consider yourselves to be rules gurus, I could also suggest: "Don't use any cards printed in the pre-Modern card frame." If you neither rules experts nor ancient (I am both
), those old cards can be fairly tough to figure out the modern templating. Also many creature cards have received updated/errata'ed types since then.
Another way to help level the playing is by modifying your build rules. For people with small collections, I often suggest: build like it is a pre-con. These usually have fixed amounts of rares and uncommons. I could suggest a limit of 4 rares and a limit of 12 uncommons. You could also modify the 4-of rule. Perhaps only allow a maximum of 3 copies of a card.
Bottom line is: If it is just the two of you, do what works for you.
Happy Brewing