Pretty much what the title says. I would like to install a 3rd party app on my iOS 11 device, but the only way to do so is by self-signing the .ipa and installing it via Cydia Impactor.
What are the possible security concerns in doing so?
How much control and access would said app have over my device once trusted?
There's no way of saying for certain.
iOS 11 fixed a number of vulnerabilities that allowed access to personal info such as SMS messages without user knowing. There might be other vulnerabilities but it looks like iOS 11 is pretty good in regards to private API access. Pretty much every known personal info leak was fixed.
The problem here is you can't do anything about it. Only way is to disassemble the application and see for yourself.
Of course, there're many ways third-party app can steal your personal info if you give it the access. For example, contacts, calendar, call history (without phone number but still), microphone, photo library - once given permission to access, can be accessed at any point even when running in the background without you knowing it. Application can run in the background indefinitely, it's still not fixed by Apple.
Accessing location will always display an icon in the status bar so that's fine. But there're ways you can fool iOS to access location once and not display the icon. It's not reliable but it works.
If application is running in the foreground then it can access camera, microphone, location and iOS will not tell you anything about it. Recently there was an article about camera specifically. You can't tell when it's recording and that's a real problem.
Solution here is simple - don't give the permission.
Jailbreaking relies on a number of vulnerabilities to modify kernel to disable security measures. No one is stopping you from using these exploits to be executed from a third-party app not meant for jailbreaking. In fact, all recent jailbreaks are done by installing an application using Cydia Impactor. And there's an actual example of that - Filza. It's a file manager that can access root file system. It does that by executing exploits used in jailbreak.
Solution here is to make sure your iOS is up to date and has no jailbreak for it. Of course there's a possibility of unknown exploits. Serious vulnerabilities found regularly in iOS. You can disassemble the application to see what it's actually doing but I don't think you would asking this question if you knew how do that.