There's a humourous saying that British and American people are "divided by a common language". It's true that there are some words and idioms that are unique to each dialect, but we understood each other perfectly well. Most of the English-speaking world is familiar with American idioms through US movies and TV shows being widely syndicated. As for the other way around - my experience in the USA as a British person has mainly been that certain things I've said have been noticed as being "typically British", but rarely have I not been understood. When confusion has arisen it is usually over terms for things, not idioms or sayings.
Your example of 'beat around the bush' is possibly a misunderstanding on your part - that note in your dictionary is pointing out an alternate UK variation, not saying that the idiom itself is uniquely UK English. This ngram shows its use in US English publications only.
As for whether you will be "laughed at" for using idioms from other dialects - I presume you are not a native English speaker anyway. There are probably lots of analogies, metaphors and idioms you might use from your own native language while speaking English anyway, so using a British one is hardly going to stand out.