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Doing the work outside of Upwork is against the TOS - that is, if they are asking you to work (and be paid) away from Upwork, then that is violating the TOS and you will also be in trouble if you accept.
Communicating outside Upwork is a bit different - the Upwork messaging system is not great, and lots of people prefer to communicate away from it. Or, they might want to do a Skype interview or something. Those things are okay. But, especially with a new client you haven't worked with before, you should keep the bulk of the communication on Upwork. That way you have a clear record of what was discussed, in case any problems come up later.
Please take screenshots of the client's request to take the job or process a payment for your work off Upwork, and contact Customer Support. Offering and accepting payments outside Upwork is a violation of Upwork ToS and carries a penalty for all parties involved. It also means forfeiting the payment protection Upwork Protection Programs provide.
Regarding direct communication, it's fine to communicate with your clients by other means outside the platform after they respond to your proposal. I do however advise you to copy all the files and communication into the contract's message room after each conversation, and ask your client for acknowledgement, in order to keep a record in case a problem arises.
It's important to note that you should never start working without an active contract, which would show in your "My Jobs" tab, and should always submit your work through Upwork, even if you have already sent the files/updates to the client through email/Skype etc.