Is this always the case for words that end in して similar to how です is pronounced "des"?
From what I hear, usually, the じ in はじめまして is included in the greeting. People shorten the "shi" into sounding like "sh" is because it would be more convenient in conversations. The same applies to "desu". If you are using Romanization(changing Japanese symbols into English pronunciations), keep in mind that in order to speak to the same dialect of a Japanese local, you should shorten words like "desu" to sounding like "des" to make more efficient.