I agree with @dude. By now, in 2019, randy –with almost the same meaning* as geil and scharf– has become quite wide-spread in the US and Canada, although it still has a British ring to it. I would say that horny remains typically American and randy remains typically British. In meaning and usage, randy and horny are completely synonymous.
*However, note that, unlike German geil, English horny and randy can only used in reference to humans –or animals– which are currently or are generally thought of as easily becoming, 'turned on', i.e. in a heightened state of sexual arousal.
So, both terms refer to a living creature's state of arousal, not to something seeming arousing to the observer. Therefore, neither horny nor randy can be said of situations, films, photographs, etc.
Furthermore, whereas by extension modern German geil can be said of activities, situations, photography, music or other things that are stimulating or exciting a non-sexual sense, horny or randy cannot used in this way.
As far as I can think, the closest sex-related English terms that can be used in this extended sense would be hot or sexy.
I concur with @dude that Randy –often as a nickname for Randolph– is a commonly-heard first name in the U.S. and so I think that people rarely immediately obsess on some idea of 'horny' when they hear it.
Those who go by the nickname Andy or Andie, do, however, inevitably get chided with the 'Randy-Andy' moniker, but affectionately so.
On the other hand, the introduction of a person with the family name 'Horny' or 'Horney' might elicit some hard-to-suppress titters and the inevitable asides. There was such a girl with whom I went to high school. And of course, horny is what teenagers are 24-7.