There are several ways on how you can help fight the private servers and knock 'em out of existence. These include:

Reporting accounts that advertise:
"Ohh, I'm too lazy. The mods won't do jack to stop this anyways. They're lazy, too."

Really? You think the mods are lazy just because you didn't get justice when you reported the shaman for icing you? (which really isn't a reportable offense in my honest opinion). Mods do an admirable job handling reports and banning cheaters/hackers/scammers/you-name-it. Just because a moderator flat out ignores you when there's a hacker in the room, doesn't mean they're just sitting there all laid-back in a chair drinking iced tea. Report report report. It will make a difference. Remember, the first step to actually stopping something is notifying the higher-ups.

"But they're not harassing anyone by innocently whispering?"
Call me soft-hearted, but it really is harassment when someone is trying to get you pulled from an authentic game in to a non-official virus-planted private server.

Ignoring players that advertise:
If you are one of the few who labels themselves as lazy potatoes, then the least you could do is /ignore the player trying to phish you. Don't tempt yourself into copy-pasting that ever-so-suspicious link. Ignore and act like it never happened.

On an unrelated note, DON'T ever copy-paste the whispered message onto the main chat. It just gives the private server advertiser free publicity.

"Be a hero?":
If you're one of those techies that have a life goal of shutting down these "cybercriminals" (for lack of a better word), then by all means, go slice-and-dice those private servers to pieces. Just be aware that you must have a lot of experience in computer logistics/analytics/programming and maybe some 'connections' to pull off the job. This really isn't for the average joe.

Feel free to suggest any more possible solutions and I may add 'em up on here, and/or to also dispute any of the above solutions.

Glad you asked! Yeah... why? Well, I'll tell you why in the following paragraphs and long walls of text. (just kidding it won't be that long...)

As you may have known, Atelier801 is an Indie company that had its fair share of economic problems in the past, and probably still has today. Transformice is essentially the only way your money flows from your hands into the palms of Tigrounette (don't try to picture that, please)and the team. Every single dollar/peso/rupee/Picarat, etc. counts towards the welfare of the the France-based company.

If these private servers lure some of the Transformice population, then one or more of the following might/will happen:
Loss of potential revenue: Players = money. If these fake Transformice platforms lure someone with 10,000 Fraises, then that someone would be less inclined to purchase microtransactions with the real Transformice, since the satisfaction gained by accessing all those dream furs would be lost when (and even if) the player returns to the authentic version.

Basically, this principle can be summed up in a piece of sample dialogue:
"Why need to buy the Leopard fur when I can play with it for free at [insert private server name here]?"

Less morale for the Atelier801 team: If you have ever noticed a trend with all of Atelier801's game, you'd see that updates and progress have remained minimal. That's because there's less players to cater to and the opportunity cost of catering to many outweighs the few. It's just common sense. Would you rather impress the three prettiest girls on the track team that have an eye for you or your two "friends" that could care less for you?

But Cookie, this doesn't explain anything...

Shhhhhhhh. If there's less people that stay loyal to Transformice because of the private servers, then how would Atelier801 respond to that? Let's take a prime example of Fortoresse, a game that used to be populated and second in terms of sustained player populations among the A801 games. Maps were released, perks were added, weapons had some balancing (and misbalancing), and cosmetic choices increased. Now Fortoresse's downfall wasn't private servers, but the game population dwindled once Nekodancer came around (not saying ND was the cause). The Devs promised them this and that, and now years later, some (as a generous term because it's more like most) promises are still not implemented. In short, the Devs gave up on Fortoresse and stuck to Transformice, their moneymaker. If Fortoresse were to maintain its player populations, then A801 might have two sources of income.