On Thursday, Instagram enacted a new function when it comes to using your Instagram Stories.
According to CNET, if a friend tags you in their story, you will now be able to share that story, add it as a sticker, or add creativity with other graphics. It will also then link back to the original account that you shared it from.
The important part of this: You must be tagged in the post for this to happen.
So, in that case, you don't have to worry about your personal stories being shared - but if you have a public account versus a private account, the stories are still able to been seen by whoever follows the public account.
Instagram adds ability to share other people's story mentions of you into your own story https://t.co/YpjQ8Piiow