I once heard from my friend that the two words: 橋 and 箸 have different stress (stress falling on the first/second syllable).
Is that true?
Many dictionaries (even monolingual ones) do not show accents, but of course there is something called "standard accent of Japanese" which you should generally respect.
The most authoritative source of the standard accent of Japanese words is probably 日本語アクセント辞典 published by NHK, but there are also some online free accent dictionaries:
Maybe you can also visit Google Translate, which also has a voice synthesizer. It's far from perfect, but most of the time it should work fine with easy phrases. Apparently if you give it a context (e.g., 橋を渡ります。箸を使って食べます。) it can read them more fluently.
箸 and 橋 are pronounced in the same way.
Some people may pronounce differently due to accent or regional dialects.
But officially there is no difference in pronunciation.
Pronunciation is a very tricky question because there is no absolutely correct pronunciation.
Due to accent and regional dialects different people may pronounce in different ways.
What most people think as Japanese language is actually the Tokyo dialect.
That is why dictionaries do not record the pronunciation of words beyond the hiragana.
According to the dictionary 箸 and 橋 should be pronounced equally as はし. Any difference beyond that is due to regional accent.
The same thing happens with English. There is no way to determine the absolutely correct pronunciation. Americans, British and Austalians may pronounce in different ways.
Unless of course you want to learn a specific accent. In that case you have to pay attention to how words are pronounced in that accent.