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This question about accepting cash in AirBnB starts to make me wonder if it is possibly against their terms and conditions, and in that case is subject to be a removal of an account or any other undesirable actions by AirBnB, to exchange a messaging app (WeChat, WhatsApp, etc...) and contact the host there.

I usually exchange the account with the host, and in many cases it is the host who asks me to contact there. The response speed and overall UI/UX is far better in these messaging apps.

But sometimes the host and/or the guest may contract within the apps, and it could be concluded without AirBnB. In fact if you try to exchange your handle, AirBnB warns you in the chat box with something like "We encourage you to chat here in order to protect you." and automatically erase the handle with "HIDDEN CONTENTS SHOWN" message, though it is obvious that their implicit purpose is to assure collecting the hefty service charge.

So do they explicitly state anything about exchanging the conversation outside of their control (before staying) or is there any reported case it led to any undesirable end?

Airbnb does take a hard line against circumventing their payment system (thus denying them their fees). From their terms:

In connection with your use of the Site, Application, Services and Collective Content, you may not and you agree that you will not:

...

use the Site, Application, Services or Collective Content to find a Host or Guest and then complete a Booking of an Accommodation independent of the Site, Application or Services, in order to circumvent the obligation to pay any Service Fees related to Airbnb’s provision of the Services or for any other reasons;

...

accept or make a payment for Accommodation Fees outside Airbnb Payments. If you do so, you acknowledge and agree that you: (i) would be in breach of these Terms; (ii) accept all risks and responsibility for such payment, and (iii) hold Airbnb harmless from any liability for such payment.

Airbnb has the right to investigate and prosecute violations of any of the above to the fullest extent of the law. In addition, and as set in these Terms, Airbnb may take a range of actions against you, including but not limited to removing or disabling access to any or all of your Member Content or deactivating or canceling your Listing(s) or Airbnb Account, for a violation of this Section or these Terms.

They encourage guests to report such attempts to them. They do note that, in some cases, you might need to pay local taxes directly to the host though, so presumably they don't object if that is the case.

These rules prohibit circumventing Airbnb's booking mechanism, but as I read it, doesn't strictly prohibit all off-site communications if you still plan to book through the site. However, communicating outside of Airbnb has some risks.

In particular, you could be subject to a nasty phishing scam involving a fake Airbnb website. Such communications could lead to links that look like the real Airbnb website, but actually point to a decoy website where your personal information can be stolen. There are other types of scams on Airbnb as well, and sharing contact information to communicate outside the platform makes you more vulnerable.

Communicating outside of Airbnb also may make it harder to take advantage of their guarantees. If you exchange messages through the site, they will have a record of exactly what was promised, which gives you a paper trail if something goes wrong. If you talk off-site, you'll have no proof of your agreement if things go south. For important details like whether your room will have a lock, I'd want to have a record of that promise in case I need to use it to cancel my reservation and get my money back.

After paying, Airbnb usually gives you direct contact information, as you'll need a way to organize logistics and an emergency contact.

After paying you are free to do whatever, that's why they provide you the host's phone number. The message box is no longer filtered as well and you might as well switch to Whatsapp/Telegram/Facebook/whatever. The only exception is if you try to arrange a few more nights without paying AirBnb - a common trick is to pay for just 1 night and then pay for a few more in cash on arrival. This will get the host penalized if caught.

Before paying is a different situation - AirBnb tries very hard to prevent you from exchanging any useful contact information. However if you are successful they generally don't care - with one host we've tried around a dozen of semi-encrypted messages to get the information across (all filtered out) and AirBnb didn't get involved. That particular host also told she tries to exchange contact information all the time (to allow guests to ask about the property by phone) and AirBnb didn't try to penalize her.