It all depends on the company licensing the series. I feel like nowadays censorship in English dubs is very uncommon, at least outside of things like Pokemon perhaps, but I don't follow the Pokemon anime (haven't for 18 years holy shit) so perhaps it's gotten better about that. Generally speaking, the worst you get out of translations are maybe mistranslations that might miss the point, or dialogue that gets edited to fit the audience better; not in a censored way, just cultural references/jokes/gags Western audiences probably wouldn't understand.
For your One Piece example, it's actually been through two English localizations. The first was done by the notorious 4Kids, who had a monopoly on children's anime for a time. They were the kings of censorship and editing, writing out deaths, painting over graves to enforce this, as well as blood or even an extended arm holding a gun. Sometimes, just removing entire episodes for "questionable content." Then they edited dialogue, some story arcs, inserted tons of dumb gags, and in general they were just a really bad licensing/localization company. This applied to One Piece in probably a similar vein to what you're describing. The only real legacy the 4Kids version has is this opening, which is generally regarded as "so bad it's good."
Then 4Kids lost the license and in came Funimation to pick it up. They did things over. They removed the censorship. They kept the story intact. Fans in general were happy about this version.
Now Funimation seems to be dominating the English localization industry along with Aniplex. Neither of these do any censorship, though in my experience their main difference is that Funimation tends to "spruce up" the translation a bit to be more relatable to English-speaking audiences, while Aniplex tends to translate things more literally; at least, in their subtitles. I haven't really compared many of either company's shows between subs and dubs. I'd been meaning to watch Kill la Kill and My Hero Academia dubbed, so that would give me a frame of reference; they're by Aniplex and Funimation, respectively.
Basically, unless it's a show airing on children's television, you're not likely to run into glaring censorship issues in English dubs of anime, at least of recent years. Going back into the 90s and beyond, perhaps, but anime translations have been treated a lot more seriously lately, to the point your biggest issue is whether or not the English voice-acting will be any good.

