"I think everything begins with communication."
Translation:Ik geloof dat alles met communicatie begint.
True, but in these kind of sentences geloof is used in a broader sense in Dutch, not just religious or something that cannot be checked, but also things like: Ik geloof dat je je linkerschoen aan je rechtervoet hebt aangedaan. = I think you put your left shoe on your right foot. Which clearly is something that can easily be checked, so I don't think believing is used in English in these kind of sentences.
Believe is absolutely fine in that sort of sentence, it would just be a more disparaging way of saying it [in my opinion]. In your example; 'I believe you have put your left shoe on your right foot' might be used by a less likeable person while they are disagreeing with you in order to discredit you by making you look dumb. That sounds like a very childish example though.
People will interject mid-conversation to correct a fact all the time with "Actually, I believe that is...". They just tend not to be very likeable people.
I lost track of why it felt so important to type this out mid paragraph. Sorry...