The answer is no. In December 2016 Instagram offered users the opportunity to save pictures, photos and videos. Millions of people are trying to figure out how this works. From our testing, it is the case that no one else can see your saved pictures but you. Even if you have a completely public account, you are the only person that can see saved pictures.
This will allow you to sleep easy at night if you have saved a ton of pictures that you do not want anyone else to know about. As of December 14th, 2016 there are no public notifications that someone has saved a specific picture. This will be something to look out for in the near future as Instagram is sure to tinker with this option. In fact, it might be very similar to Pinterest in that you can create specific files and boards of photos that you save.
It would only make sense that you could file the pictures you save in specific areas or tag them with the title or keywords so you will be able to search them later.