So... Let me preface this by disclaiming one thing: I AM NOT A LAWYER. As far as I know, there's not a true "closed species". You can trademark a name. You can copyright a character (though even that is in dispute in some places), so others can't legally claim to be/own that specific character. But you can't copyright something as generic as a species without going through proper copyright channels. The ownership of such things is heavily disputed, and before you go making such claims, please consult a proper copyright/trademark lawyer about the subject. Again, this is all things from my understanding of the situation.
With that said, I've also come to learn just how difficult life can be for creators when a fanbase starts to establish itself. There are a lot of great fandoms out there, but they also have their really cruel and controlling sides to them. Steven Universe and Dream Daddy have shown this numerous times, as have the My Little Pony: Friendship is Magic and Pokémon fandoms. From people trying to shoehorn their own ideas as canon (Pokémon fandom across the generations), to attempting to push their ideas as if they're a stance of the creator (Dream Daddy genderbend incident, Steven Universe fanartist being attacked), to fans trying to outright hijack and hack their way into finding out information ahead of time (My Little Pony Season 8/9 and Generation 5 leaks). Fans are great, but some fans can be outright assholes.
This is just how people are in general, but as of late--especially on social media such as Twitter and Facebook--some fans go above and beyond with outright theft of characters and settings out in the public eye.
…Now, before I continue, let me clarify. There's a rather large difference between using an established character for fan art and fan fiction, using an established character for private/personal role play in chat or forums, and using an established character for role play in public locations such as social media. When in use for personal fiction or role play, you aren't representing that character or person in the public light. When doing so publicly over Twitter, even a parody disclaimer can be missed, and at that point, you can appear to represent that character or person in a public manner. The latter isn't good, and can lead to impersonation or IP infringement accusations.
Using Pokémon as an example, those who play generic characters such as random Feraligatr and Charizard on Twitter are in the clear, as they're OCs (original characters). But for people who play as someone like Yellow from Pokémon Adventures, or Arron's Lucario from the eighth movie, "Lucario and the Mystery of Mew"... You guys aren't in such clear waters. As established characters (ECs) with canonical backstories, people stumbling across such accounts will make assumptions, which can cause confusion regarding the "official status" of the account. This is especially true of someone under the age of adulthood stumbling upon an account of a character that's supposed to be underage, doing things they really legally shouldn't. (Yes, this actually happens, and has happened in the past...multiple times.)
This is why it's so important to make sure you show that you're not playing as the official character, but as a parody or alternate universe variant of them.
Now, there's also another thing to take into account: personal characters, or personas. Whether they're called Pokésonas, Ponysonas, dragonsonas, Digisonas, fursonas, or any number of other "*sona" variant, they are very different from ECs and OCs. These characters are most often very personal to their creators, and tied to the person behind them. For the most part, they should be considered completely off limits without explicit permission from the creator, which would be posted somewhere to be referenced.
By using such *sonas without such permission, you are effectively attempting to represent the person who created them. This is outright theft of property, and can even be construed as identity theft to some degree. And people are actually doing this on the likes of Twitter and F-List, both with permission and without.
I've seen people outright claiming to be someone's personal fursona, using commissioned art (which violates the copyright of the original artist and character owner) to go the full distance. And it confused some people enough that they went to the character's owner, thinking that it was them that they had been talking to, when in reality it had been an impersonator. And that's not good for anyone.
If you want to do such a thing, talk with the person first. If they say it's okay, ask them to put it down in writing somehow so you can host the file, and that they can be contacted in the case of a dispute. Chances are, they're going to say no, and you'll need to leave it at that. Pestering them about it won't help your case.
In the end, the best advice to be given is this: Make your own characters. It's not hard to come up with a name (it doesn't even have to be a really unique name), a small backstory, and some small details about what makes them different from others of their kind. The more elaborate characters will be more difficult to make, but in the end, make a character you want. If someone calls said character a "mary sue", ask them what they'd do different, and see if they'd help you. If not, just tell them to put their skills where their opinion lies, or ignore them and be happier for it.
Make something you can control, instead of using someone else's property. You'd be surprised at just how powerful such a thing can make you feel.