Can a thief use my stolen car registration?
I accidentally left my car unlocked, and it was searched by trespassers in the night. They only took the registration and insurance card (Ohio, United States). This seems strange to me, as these things aren't really useful to anyone, except possibly for the personal info they contain, although I don't think it's enough information to take over online accounts or to impersonate me in person. Strangely, they didn't take low stakes items like change, or high value items like a dash cam and expensive sunglasses, or even the actual plates, which would have been useful along with the registration.
What should I be worried about as a result of this info leak?
It's hard to give an exhaustive list, but in the end, I don't think that's really the important question to ask - really, the important question is, what should you do now to protect yourself?
Reach out to the DMV and your insurer. Your state's DMV will have a process on how to replace stolen car registration documents. The risk of not reporting them as stolen is that the thief may try to impersonate you with respect to your motor vehicle ownership (i.e. they may try to create an online account with the state DMV, or take over your existing account. Or they may try to get replacement plates and impersonate your vehicle, etc).
Ask the major credit bureaus to lock your credit report. This will mean a minor hassle for you if you try to apply for credit any time soon, but it can help protect you from identity theft. The thieves now have your name and address and details about a vehicle you own - while that by itself isn't very powerful information, it may form a piece in a larger puzzle - taking steps to protect your identity can help prevent that.
My Car Registration Was Stolen. Should I Be Worried? - common reasons for registration theft
Obtaining the Vehicle Identification Number (VIN). Gaining access to your VIN may help thieves register illegal vehicles. They could also file claims on totaled cars or even create duplicate car keys. There have even been instances where individuals use a stolen car registration to walk into a dealership and drive off with a brand-new vehicle.
They're preparing for a break-in. Your car registration contains vital information, including your home address. Depending on what the thieves find in your car, they may target additional valuables kept at your residence. Although this specific scenario is less likely to occur, be sure to take extra security precautions in the days, weeks, and months following a vehicle theft or burglary