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This user page is about the capitalization of the "common noun" component of proper names. Wikipedia:Naming conventions (capitalization) says Do not capitalize the second or subsequent words in an article title, unless the title is a proper name. Wikipedia:Manual of Style/Capital letters#Proper names says that proper names are usually capitalized. So barrier island would be lowercase, since that is a descriptor of a type of island, but Berrier Island would be uppercase since Berrier is the name of a specific island.

If there are multiple proper names that begin or end in the same word, though, that is when things get tricky. I have not found guidance on this in the MOS, so here is what I usually do. For multiple nouns which are referred to as a group, such as Long Island and Staten Island, I would lowercase the common-noun portion of the names, per the AP Stylebook (for example). Such as between Long and Staten islands. The conjunctions "and" or "or" would be a good indicator of this. The AP Stylebook says, Exception: plurals of formal titles with full names are capitalized: Presidents Jimmy Carter and Gerald R. Ford.

For multiple nouns that are referred to as single entities, I would keep them uppercase: from Long to Staten Islands. This applies to common nouns like streets, rivers, bridges, and buildings. I still haven't figured out what to do for multi-word nouns like police departments.