The fact that we refer to the different ways chinese people speak as dialects is a little confusing. Actually, the chinese dialects are different languages just as much as, say, English, Dutch, German, French, Spanish, Italian, Portuguese, etc.. So no, they are not mutually intelligible, though they may contain many similarities.
The situation with the writing is a bit different. In fact, only Mandarin has a standard writing system, so no matter which part of China a person comes from, or which dialect is their mother tongue, local publications will be in Mandarin. Therefore, to be able read this material, one must be literate in Mandarin, even if one cannot speak it! Most official publications in Hong Kong are also in Mandarin. However, chinese characters have been adapted in order to write Cantonese, and so some things may be writting in Cantonese. However, under these circumstances, cantonese grammar and vocabulary is used, and therefore may make as much sense to a Mandarin speaker, as say reading spanish would to a french speaker (in other words, they may be able to understand the general meaning if not every word).