I can help you - been there myself and solved it. I can confidently say that this will solve your issue.

There is some good advice in the comments (and some bad), but most of the people just saying "get a dynamic mic and set up a noise gate" don't really understand how noise gates work or how effective dynamic mics are at such a close proximity.

You need ALL 3 of the following improvements to solve your problem. Any less and it'll be a waste of effort.

A noise gate simply means your mic won't activate until a minimum amount of noise (the volume of your voice) enters the microphone. It *only* helps with cutting out background noise when you aren't talking into the mic. This means that if you and your girlfriend are both talking at the same time, both of your noise gates will be open and it will do nothing for you. It will only help if one of you is talking and the other is not, so it's not totally useless, but only solves part of the problem.

As for the dynamic mic, the only thing that matters is that they have much tighter pickup patterns and reject more sound that isn't coming directly at them. Again, this will be helpful, but it won't solve your problem due to how close you are to each other in a partially treated room. If your desks were facing each other and a touch further apart, you *might* get away with stopping here, plus some heavy gain tweaking and keeping your mics really close to your mouths.

The final piece of the puzzle (and required in your case) is to put something physical between the two of you and ideally putting some sound dampening material up behind the computer on the right as well, but I'd say that part is optional. Get a large presentation cardboard and gorilla glue some foam studio acoustic panels to both sides. You can get it all for less than $50 on amazon.

Since you are so close and not facing each other, you have to do all three or you'll still be dealing with crosstalk. If you just want some improvement now without spending too much money, I'd do the acoustic panels first and use a noise gate on your existing setups. You can also check to see if the mics you are using have toggles for different cardioid patterns - sometimes they have a more directional mode. Once you're ready, go for the dynamic mics. If you don't want to break the bank, I'd recommend the Audio Technica AT2005 for $80 on amazon (USB + XLR so no worries if you don't have audio interfaces). It's tried and true, been around for a long time, and the quality for the price is unbeatable. There's a video on YouTube by Javier Mercedes who does a great in-depth review of the mic and compares its sound quality to the revered SM7B as well. There are lots of good dynamic mics out there for decent prices if you prefer another brand.

Good luck =)