Let me use an example to explain how this works:
Player A activates the effect of "Drytron Alpha Thuban", tributing "Cyberangel Benten" for cost. Player B chains their set "Jar of Greed". The chain resolves and Player B draws "Droll & Lock Bird" off their "Jar of Greed". Player A resolves "Drytron Alpha Thuban", summoning it in DEF Position and searching for "Cyberangel Natasha". On resolution of the chain, Player A activates the effect of "Cyberangel Benten" that was tributed, to search for a Fairy Monster. Player B CAN in this instance now chain "Droll & Lock Bird", thus preventing the Search off of "Benten's" effect.
If you meant it like this:
Player A activates the effect of "Drytron Alpha Thuban", tributing "Cyberangel Natasha" for cost. Player B chains their set "Jar of Greed". The chain resolves and Player B draws "Droll & Lock Bird" off their "Jar of Greed". Player A resolves "Drytron Alpha Thuban", summoning it in DEF Position and searching for "Cyberangel Benten". Player A then proceeds into an open gamestate and as turn player activates "Cyber Emergency". In this instance Player B CANNOT chain "Droll & Lock Bird", because "Droll & Lock Bird" is also a trigger effect that can only be activated AFTER resolving a chain in which your opponent added cards from their Main Deck to their hand.
I hope this clears everything up