The criterion, in almost every case, is residency in the U.S. Most offshore brokers who refuse U.S. clients say specifically that they do not accept "residents" of the U.S.
Pepperstone, which you mentioned, does not use the word "resident", but their policy has the same effect. Here is what they say:
"Pepperstone is not able to accept applications from applicants in the following countries: United States of America, New Zealand, Japan, Burma, Canada, Côte d’Ivoire, Democratic People's Republic of North Korea, Democratic Republic of the Congo, Eritrea, Former Federal Republic of Yugoslavia, Iran, Iraq, Lebanon, Liberia, Libya, Somalia, Sudan, Syria, and Zimbabwe."
Over the years, we have vetted a long list of offshore brokers, including a long list from Australia, who will not deal with us. On post #3 of this thread, we list them. Here are the Australian brokers we have looked at -- who will not accept us.
Ava Capital Markets Australia PTY Ltd (Australia)
Axi Trader (Australia)
Berndale Capital (Australia)
CLM - Core Liquidity Markets (Australia)
Direct FX Trading PTY LTD (Australia)
ForexCT PTY Ltd (Australia)
Forex FS (Australia)
Forex Nation (Australia)
Global Prime (Australia)
GO Markets (Australia)
Got Money FX (Australia)
Halifax Investment Services (Australia)
IC Markets (Australia)
IG Markets (Australia)
Kinetic Securities (Australia)
MXT Global PTY Ltd (Australia) --- see Vantage FX PTY Ltd (Australia)
Vantage FX PTY Ltd (Australia)
Expatriate Americans, who retain U.S. citizenship but reside overseas, face different -- but, equally difficult -- issues in trying to do banking and trading with offshore banks and brokers. All this is thanks to the U.S. Nanny State, and specifically to the FATCA regulations (which I won't get into in this post).