Yes, silver axe appears on our own island.
What a few posters have said - shop for the items & then go, so you are on the island for probably around a minute - so called island hopping
- this is what I've always done, and other players really don't like it. It has been the reason why I've had some trouble on CT islands (I get blocked very fast, before I can leave, all they have to do is open their inventory).
All you can do then is either wait it out or error (& sometimes the other player errors). So you lose medals and the items.
I never engage with other players, mainly out of shyness, not trying to be rude, - that in itself has annoyed some of them enough to error on me.
Reason why island hopping isn't liked is that the game saves for everyone when you enter & also when you leave. I can see that it is annoying. (stops you playing).
Haven't been for years & with all the hacking doubtful I'll go now, I can forgo the exclusives, not worth it.