Seeing that you have arrived, I can safely assume you know what are puffy nipples, and how are they related to Gynecomastia. If I'm wrong, please checkout this article
Now that I know, you are familiar with your condition, lets deep dive if this condition requires medical interference or it would go away on its own. Please know that information is relative to my condition and to what I have read all across the internet and cannot compete with getting an actual doctor's opinion.
I started getting puffy nipples, when I was around the age of 10 and it was devasting particularly as I was a big kid with big man boobs and nipples as well. Never removed the shirt in public, constantly pinching the nipples to make them smaller even for a little while and all that stuff.
For reference, these are how my nipples look like at the age of 22.
For me, this condition has stayed all my life even when I reduced my weight to a normal range and even when I was overweight. It didn't matter if I could bench 130lbs but this condition never went. When I was 15, I attributed this to puberty being a bitch and didn't go to the doctor hoping and praying this condition would go away on it's own but that never happened.
Again, I repeat this is according to me, and you might differ. It would be best if you could show this to your doctor and get his opinion on it before you come to any conclusions but from what I researched and read all across the internet, this condition mostly will not go away on it's own and it would be best to get surgery. The cost would differ according to the country but in a place like India it would come around 2500 USD.