Ila, Your doctor wouldn't test for that unless you asked for it specifically, and had good reasons for wanting to know. You'd probably have to talk him into it. I'm pretty sure those tests are more expensive than just doing the A1c and blood glucose that they usually order.
If you eat 15 carbs on an empty stomach in the morning, and skip your insulin dose, and your blood sugar doesn't rise dramatically and stay high, then you probably have some beta cell function left.
Since both of you started insulin fairly recently, your beta function is obviously on the decline, but it takes time for it to burn out completely. If you hadn't started insulin, it would have continued to decline as your blood sugar levels rose. The toxic effect of glucose on the beta cells increases as your blood sugar goes up, and the longer it stays that way, the worse the damage becomes.
I've been on insulin for a few years now. I had to start it at diagnosis; my blood sugar was through the roof and I was terribly ill. My doc said at the time that she couldn't figure out why I was still breathing. My beta cell function was pretty far gone by that time.
Stemwinder, you started insulin recently too, so you likely have a fair amount of beta cell function left.
Of course we all know that the more careful we are to keep our levels under 170 after meals, the better off we'll be long term.