I tried to search the Register button on the site but could only find the Sign In Option. Does this mean that the registration on the site is already private and only available for the bitcointalk members.
There's no option to register directly trough the website. You need to send a message over to joker_josue if you're interested in signing up for it[1].
If that is the case, then it is good that the site is exclusively built for bitcointalk members but anyone can upload the images on talkimg as sign in is not required to upload any image. Do we need restrictions on that part too?
Per joker_josue reply before, images that are posted outside the forum will eventually stop working because the service wasn't made to be used outside bitcointalk (thus avoiding unecessary burdon on the bandwidth and disk space). By registering you will be getting a private API which you can use, for example, to automatically upload images using programs such as ShareX (more information for that specific program here[2]).
[1]
https://bitcointalk.org/index.php?topic=5450546.msg62329206#msg62329206[2]
https://bitcointalk.org/index.php?topic=5450546.msg62281428#msg62281428