Legally, the requirements about what you collect and how you notify your customers differs from country to country. See a lawyer about that.
The data we collect (the computer "fingerprint") is anonymous, and useless outside the context of activating your software. In other words, everything that LimeLM / TurboActivate creates and uses is both anonymous and useful only to LimeLM -- the data cannot be used to track back to any particular user. Any other data you put into LimeLM (customer's email, name, etc.) it's entirely up to you to comply with the appropriate data / privacy laws of your City / State / Country.
Is it necessary to have a privacy statement?
It depends on the laws of the country you're in and the countries you'll be selling to. See a lawyer.
As a rule of thumb, yes, always tell your users what you collect and how it's used. Savvy users will avoid software that doesn't explicitly tell them this.