Usually you don't use this. Times that you use this are (as far as I know) when you are being extremely polite (like talking to a king or in prayer, basically almost never) or when you want to fight someone.
Like the others are saying pls don't use 당신 for like anything it's rude in most cases. Also 너 is also rude! In Korean you don't say 'you' very often even in 반말, but say people's names and their titles (오빠 언니 등)
I'm pretty sure it is supposed to sound like "dangshin" because if a characters last consonant/the character ends in "ㅇ" it is pronounced as a "ng" sound. E.g. "안녕" is pronounced as "annyoeong". Hope this helps :3