Saving a picture usually isn't a problem, but some app developers may make it more difficult for you to view that saved media using the phone's usual gallery. Instead, they might make it so that you have to stay within that app to view the picture. This could be beneficial for the app developer, because the more time you spend in that app, the more data they might be able to collect from you (and many companies profit from that kind of data).
Install a file manager like
Total Commander (which is free and ad-free). When you open it, tap "SD-card" at the top--this actually represents the phone's main internal storage, and not a microSD card. Now start tapping some of the folders you see, and look for something that might be related to Kik. The folder might have Kik in its label, but this isn't always the case. I'd focus on the following folders: Android, Media, and Pictures. Each folder may have more subfolders, so you can keep drilling down. To go back up a few levels, tap the bent arrow pointing up at the top of the screen (or tap the blue Home button at the top of the screen to get back to the initial screen).
If you do find a folder that is specific for Kik, then look in there to see if there is a file called ".nomedia".