As described above, on federal land where the enforcement agency has either concurrent or exclusive jurisdiction, the federal agency can assimilate state laws as long as they don't have a federal law on point for the given offence. Most traffice laws for example, do not exist in federal law, but can be enforced on fed lands based on the code cited below.

Assimilative Crimes Act, 18 U.S.C. § 13
The Assimilative Crimes Act, 18 U.S.C. § 13, makes state law applicable to conduct occurring on lands reserved or acquired by the Federal government as provided in 18 U.S.C. § 7(3), when the act or omission is not made punishable by an enactment of Congress.

With regard to federal agencies authority on state land, it varies from state to state. Some states grant federal agencies enforcement officers deputy or peace office status which allows them to enforce some (or all) state laws on state lands. Alaska for example, offers USFWS and NPS enforcement state authority on state lands (state wildlife trooper equivalent) for fish and game regulations only. The motivation for this cross deputization is typically to support joint missions and objectives in areas of reduced resources and man power. Rural communities for example. In cases where this type of federal/state crossover exists, the office should have a credential explaining their area of jurisdiction and given authority with such.

As a rule of thumb though, federal folks work on federal land and state folks work on state land...or something like that.

Again subject matter jurisdiction is different entirely and is much less limited by land boundaries and ownership.