The most extreme example of this is Japanese Kanji, and while Japanese is its own language, native chinese speakers would note the similarities between it and Hainanese/Cantonese dialect, and with some 文言文 knowledge, one can read Japanese without much effort, even if one cannot speak it.
Contemporary Chinese has roots in the version used among the masses, it deliberately eschews the official version used in the palace to make it easier for the masses to understand and learn it. One may notice the difference between Taiwan official language and Mainland official language in official declarations, the Taiwanese one is considerably more ... "poetic" but comes at a cost of understanding for the masses who may not be educated as highly, like most of the villages in Mainland China, likewise for Hong Kong.
However, one might notice how written Chinese has relatively little "hard" grammar rules that must be followed to ensure legibility. As a translator I find it easier sometimes to translate something into Chinese, which is not my first language, and then to the target language, especially for Sinitic languages, because the written language can capture most of the meaning while being flexible enough to be restructured in the target.
The converse, on the other hand may be tougher, for example if Hong Kongers were to use Cantonese as a base to understand other dialects they will face a lot of problems in doing so because its not flexible enough to fit in the quirks of other languages, which is why in this sense written Putonghua is a successful invention because its minimal enough to convey meaning while being able to fit it all the weird quirks for various "dialects", which is why its used widely across the world.