I'm debating on which way I want to go with this. 

Depends on how you want to do it. You can either be completely up front about it and just say "Yeah, the monster uses its Legendary Resistance to make that save....." or you can just roll the dice anyway, knowing that the monster will just make the save regardless of what you roll.

If you want to make it a bit more immersive, you could just roll (or not roll) and describe how the monster resists the effect in a cool way. If you describe it cool enough, your party may not care if it was a rolled save or not.

Are you trying to hide it from your party so that they don't know the monster has said options? Or is this more of a meta-game thing where you don't want to give away when the monster uses the action so the party knows when they are in the clear to go all-out on spells?

Good question. But I personally tend to give as few details about the mechanics as possible, so no.

For this type of thing, I'd usually say no. But my players are all new to 5e. The only experience they have is some of them watch Critical Role. So for the ones who don't, and for everyone as a whole, it can be really cool/intimidating for you to say "Well he failed his saving throw, but because of the monster's XYZ, he just chooses not to succumb to the spell." 

Which I think can be a cool "WOAH!" moment for the PCs. Rather than just behind the screen saying "Yup. He saved against it."

More a debate on whether or not its more interesting for players to know whether or not the monster has used it's legendary actions to know when to use their big spells. 

I feel like either way could be fun and strategic. 

I can see for the first couple of monsters going "he starts to be taken over by the spell and then shrugs it off.." or something similar. 

Not sure if it would become repetitive at some point. 

It hasn't come up in either of the games I'm running, so I don't know if I'll be upfront about it or not. I think not, unless there's a in-game reason for one of the PCs to know about it (i.e. having a ranger in the party and going up against one of their favorite enemies). Unless the party has actively studied this creature to know as much about it as possible, there's no reason they would know about legendary resistances.

Yes. My players and I like to talk about game-mechanics, so it suits our tastes. If you're going for hardcore storymode, eh, I can see an argument for it either way.

As in you don’t describe your monster’s actions or you don’t give numbers? 

I find that combat is the place where the mechanics in D&D most show, so while I love being descriptive and immersing my players in the excitement of combat, I don't tend to obfuscate the mechanics since it becomes more engaging/fun for the players when they're involved in the mechanics. So yeah, I tell them "the dragon uses its legendary resistance" or what-have-you, even if maybe the characters wouldn't know. 

I think it's definitely a "flavour to taste" sort of thing though, each campaign and group has what will work best for them.

I generally do something to where I sprinkle the phrase in just so they know what happened but I don't give up the specifics like if it has any left or anything

"Your fireball explodes onto the giant whatever as it simply tilts its head to the side, narrowing its eyes as drool begins to drip from its slathering jaw, its legendary resistance to blah blah sending a twinge of panic down your spine"

It adjudicates the action, explains it with narrative, and lets them know I'm not just being a dick gm without giving them much else to go on. 

Quote from Hallo_im_Mike >>

As in you don’t describe your monster’s actions or you don’t give numbers? 

 I prefer to use descriptions that might hint at the Legendary Resistance trait, but, most of the time, I don't say "the monster uses Legendary Resistance".

I do, since I usually announce my saving throw rolls.

I'm fond of colorful words and using as many adjectives as my grey matter will allow me to recall when I do anything in D&D. Will I announce Legendary Resistance, no, I will give some description that indicates that the spell did not work as intended.

"You release the magical energies, a spell you've used countless times, and watch as the expected result...isn't."
"The beast roars as you finish your incantation, the miasma swirling up around it, as the creature steps forward un-phased by the incendiary cloud you just cast."

However, if there is some misunderstanding or confusion and I need to explain a mechanic, I have no problem doing so. I just wait until a break, or after the game, this way things can keep moving forward for the time being. 

had a group dissolve over this. the guy had homebrewed a creature with l legendary resistance per round regeneration.  it was stunned then passed a fireball dex check.  it got into an argument that led to 2 of the players being kicked out of the restaurant we played at because the dm didn't want to reveal the creature was using legendary resistance since he didn't want us to know it had them.  after session the other two players were told why it passed. we left the group.  so if it seems to be causing a problem i'd say reveal it. otherwise it is the dm's choice

Wow, quite the Raise Dead on this thread.

But on the topic, yes, 100%, I always tell them I am using a legendary resist. But then, I roll Saving Throws for monsters in the open, because I don't want the players to think, "oh, the DM is just letting his pet monster save vs. my awesome ability." I started doing this quite by accident when our sorcerer kept trying to Hold Person a BBEG (it turned out the creature was not humanoid and the spell shouldn't have been able to affect it, but I did not think about that during the battle, being a newish DM). I knew that Hold Person would pretty much end the battle and the players would probably assume I didn't want it to be so "easy," and might have a reason to fudge the rolls. So, to demonstrate that I was not fudging, using the VTT, I rolled the saves in the open (I kept rolling to-hit and damage with my own expensive metal dice -- I paid for them, I'm gonna use 'em). The hold failed like 3 times in a row... I just rolled high. I know if I had been rolling "behind the screen" eventually the player would have thought I was just trying to stop it from happening. I had no need to do that, since he had legendary resists, but the player did not know it.

Anyway, ever since then, I always roll every saving throw for NPCs in the open. Therefore, they will see it fail, and I would then say (I have only had to do this once or twice, ever), "He is going to use his legendary resistance to save instead of failing the save." This way they know what is happening and that I am not just fudging rolls.