For the most part, an Effect Monsters effect can only be activated while it is on the field. If the monster does not say otherwise, this is the rule.

However some effects say otherwise, and you should follow the card text when in doubt:

Some monster effects activate when they are sent to the Graveyard. "When this card is destroyed and sent to the Graveyard", the monster needs to have been on the field before it was destroyed, and it needs to actually go to the Graveyard (Banisher of the Light might prevent this). "When this card is sent to the Graveyard", the monster doesn't have to have been on the field, discarding the card (as long as you don't discard it "to activate the effect") is enough.

"Discard this card" (e.g. Kuriboh) effects can be activated while the Effect Monster is in your Hand. Exodia the Forbidden One has a Continuous Effect (the effect was active the whole time and does not start a Chain) "while" it is in your Hand.

Marie the Fallen One has an effect that is used every Standby Phase from the Graveyard - so it's not on the field either.

Some monster effects can be activated from the Deck. Red Eyes Black Metal Dragon can be Special Summoned from the Deck by Tributing a Red Eyes B. Dragon equipped with Metalmorph.