I ask for these on web forms simply because it's good to be able to reply to people by name, instead of starting your response with "Hello" or, as is sadly quite common, no greeting at all.
> I can't keep a full communication log on my side.
That bugs me too. I use a "receive a copy of this email" checkbox on my forms for this reason.
Then 1. make it optional, not mandatory, I really don't care for being greeted in an answer to one of my inquiries and "hello" more than suffices (nothing is also perfectly acceptable).
And 2. Don't make it a pair of First, Last field. If I'm Her Majesty Elizabeth the Second, by the Grace of God, of the United Kingdom of Great Britain and Northern Ireland, and of Her other Realms and Territories, Queen, Head of the Commonwealth, Defender of the Faith you'd better greet me as exactly that. On a more serious notes, various cultures have different takes on names and greetings, not everybody has a first and a last name (or only a first and a last name). Just put "name" and I'll put what I want in that.
> "hello" more than suffices (nothing is also perfectly acceptable).
To me -- as a receiver -- it feels impersonal to get a barked response without a greeting, especially if I'm writing to a company or in a professional context, and particularly if their answer seems canned. And, as a sender, it feels uncomfortable starting without a short personal greeting (at least for the first email you send). In the field of customer service, especially, you need to convey that you've acknowledged the sender's unique concerns. Even when they're far from unique.
Some interesting reading here in "Are greetings and salutations redundant in an e-mail?": http://english.stackexchange.com/questions/18396/are-greetin...
In case it ever crops up, addressing Her Majesty The Queen in written form as 'Madam' and closing with 'I have the honour to be, Madam, Your Majesty's humble and obedient servant' is just fine.
When the person who mails you is actually involved/interested, even a greetings-less mail won't sound impersonal. When the person who mails you is not a person, you probably won't get to make it personal. And in the worst case, you may land it right in the uncanny valley.