You are certainly allowed to share the news of your scholarship - many do with newspaper announcements, FB posts, even the "big check" presentation where at an award banquet or pep rally with the whole school present your young adult receives a check with like 240,000.00 or whatever mentioned by a member of that service in uniform. He'll learn and you'll all learn about personal and operational security and not posting things like "My DS is heading out to see on the USS Alexandria this summer - they are picking him up at ___ port on _____ date."(not a good idea to tell the world where a submarine will be - they're kinda sticklers about that for good reason) But stating he's training to serve - I see that as OK.
One perspective on this - I'll just caution for all readers with young adults receiving their scholarships, as I have stated here on the board and in-person in recent years, that not everyone who wins the scholarship and uses it makes it all the way to commission. Some with NROTC scholarships believe it or not don't even make it all the way through the summer NSI training before their Day 1 of Freshman year - over nearly before it starts. Some quit without penalty before the start of year 2. Others - mechanical injuries, grades, discipline issues, ability to pass 2 semesters of Calc/ Physics for Navy option, failed drug tests, one arrest/ DWI... I have empathy for the young adult who tells the world they are going to be a Navy Officer (woohoo!), takes the standing ovation and big check, receives a heroes laudation, and then in the years to come has to run into people at the supermarket or football game, and explain that didn't work out. Maybe they shouldn't have made the big announcement... I don't know. I know one of these - his military career ending "sin" if you will was getting a picture taken at a college halloween party his freshman year dressed as a certain someone with a thin mustache, short in stature, with a swastika on his arm of an SS uniform, and giving the straight-arm salute. I guess he thought it was funny- others-did-not. Party picture taken over a weekend - outrage ensued on social media, and he was drummed out of ROTC in a PRB as quickly as they could organize it. School would have done the same except his Daddy has big connections. I remember his big check ceremony too - I cringe for him when people ask how it's going...
If it further helps, I observed indirectly, mostly from stories from an EMT at NSI and from other parent posts, an inverse relation to the all-stars at JROTC, with 15 medals on their uniforms, come in overconfident their struggles who in SOME cases thrived and in some cases scrubbed out of NROTC 4 years ago. Some were the people walking around asking others "Do you need help?" getting totally overwhelmed and leaving within days. (*and others as would have been predicted kicked *ss) So it's not about past talent - certainly an ISR winner is pinned to be a top talent - I just see a benefit of keeping low key and humble until you get a little further along. You do you - just sharing one perspective that may help some young adults save face if they decide serving is ultimately not for them.
My DS is now a couple of months from commissioning as a Navy Ensign. When he does, I'll START wearing the Navy Dad apparel, but a yardarm in my front bushes, start using nautical terms as I navigate a canoe this summer... but for the last 4 years we've avoided the press, the ceremony, the celebrations until he's actually serving. Just one approach for consideration.