It all depends. First off, when it comes to the hammy stuff, just bear in mind that you're using One Piece as an example. Secondly, it all varies from situation to situation as far as that goes. Some of the Japanese Voice actors put in extra emphasis or "emotion", while some don't. Some American VAs sound more mature, while some don't...
Example where Jap>Eng:
Japanese
English
Only issue is the melodramatic dialogue...
Example where Eng>Jap
Comparisons of both...
The dialogue is a little different, but oh well...
These aren't the BEST examples, but they were the first that I could find offhand, and they make my point to an extent. Basically, Japanese VA tends to have more of a dramatic tone, while English VA tends to sound tougher, more gruff, manlier, etc.
Keep in mind these are just generalties, and are just general examples and several exceptions exist.