Your images are private unless you share the URL of each image.
You are the only one who can see your image until you share the link.
After it is shared anyone who receives the link can see it.
This is an example of a Gyazo link:
Each link (URL) is made up of a long string of random characters that can't be guessed. It is also unlisted and can't be found in your favorite search engine until shared on a public website or app.
How private is an image until I share it?
There are 2 to the 128th power (2^128) of possible unique addresses. That's 340,282,366,920,938,463,463,374,607,431,768,211,456 possibilities!
Since Gyazo is also taking measures against DoS attacks, it is impossible to figure out the ID through a large amount of access.
When you share an image with someone, they will only see the image unless you choose the option to share info. Related images can only be seen by the user who uploaded the image. For more details on this, please see this article:
Can someone else see my metadata when sharing a Gyazo image?