If somebody where to say "Io sono sicuro" to convey the idea that they are feeling safe, I would wait for them to specify of what exactly they are sure of.
A personal anecdote seems fit here. When rock climbing, it is custom for the lead climber to call out to the person belaying (securing with a rope from a lower position) him to call out "sono in sicurezza/sosta!" when he is in a confortable position to secure the belayer, i.e. he is "al sicuro". This can be quite important as the belayer occasionally does not see the climber, therefore only relies on verbal communication. An Australian friend of mine, wanting to impress us with his Italian and leading the climb, shouted "sono sicuro!". The answer from the belayer (my friend, Italian) was automatic: "Di cosa?!". Things were immediately cleared up, but all in all an amusing story.
As @DaG suggested, it goes without saying that when using "sicuro" as an adjective referring to a specific place, it is intended that the place is safe.
Questo posto è sicuro.
Questo è un posto sicuro.
As a term, it is also used in technical/journalistic jargon
[...] estendere alla Tunisia lo status di porto sicuro.
