Read Jörg Porsiel's article in Multilingual and let it sink in.
The source text has most probably been saved along with the time stamp, the IP address and the company URL of the sender on the providers server. In the same way, the internet translation is saved along with the time stamp and IP address of the receiver. In addition, it is conceivable that at least one more file is saved a file containing terms not previously known to the MT system. All of this data can be linked together easily and freely.
Well, depending on the motives and the criminal intent of the provider, it is likewise conceivable that data gathered is evaluated by special software, collected and put together according to particular search words, IP-ranges or URLs. Depending on the content of the source text, subject matters can be analyzed in conjunction with the IP address of the sender/receiver, and with relatively little effort from an IT perspective, an entire personal profile could even be created, particularly if the source text contains personal information or obtainable data such as names, addresses, e-mail addresses, company ID numbers, bank details, contract details, deadlines, payment details and currencies used for payment.
The end client is most probably aware of this, even if you are not. They ask that their data be kept safe, something that should be self-evident. They are asking for a professional service. And possibly even pay for one.
They may never find out that their request was not respected. But at tthe very least the middleman easily can. A simple search for your name (plus perhaps "Google Translate") is enough: this string on a public forum will show up even in several years' time.
Attila