If, as Yini suggests, this is rampant in e-mails, then don't expect strict adherence to rules of grammar and/or punctuation. I think a comma would work wonders, because you should be analyzing "noted with" as a unit, rather "with thanks". "Noted, with thanks." makes it clearer, I think. "(Whatever you said or did has been (by me) duly) noted, with (my) appreciation/thanks." It's not the epitome of elegant communication, but it bothers me less than it does Yini. Sorry ;)

I get this a lot at work especially on emails, frankly I don't like this phrase. "Noted with thanks" means that she has taken note of what you told/sent her and she thanks you. Grammatically I don't think it is correct and I don't know where or how it started. However it is used very commanly with emails, but not spoken. I would prefer to use "Noted. Thanks".

probably equals: with gratitude.