Either it accepts millions of dollars in federal aid to build a protective sand dune and for the first time in its nearly 90-year existence allows the public to use its beach or it rejects the aid and retains its privacy, potentially worsening an erosion problem that saw part of its main road washed out and power lines toppled during the October 2012 storm. [...] some of the 600-plus residents in the village of million-dollar homes worry opening up the area could lead to traffic problems, trespassing and more garbage. After Sandy, Congress granted the Army Corps $5.3 billion to study damaged areas and for projects to build dunes, enlarge beaches and install structures to slow sand movement. The problem: Because of its small population, it has an annual budget of just over $2 million, so Letica is asking officials to ease the public access requirement. Breezy Point, a cooperative on Queens' Rockaway Peninsula that flooded and burned during Sandy, was given a $1.2 million grant from the Federal Emergency Management Agency to study building a $57 million dune with a sea wall. Emily C.