She told the dispatcher her name was Sarah Waters

LMAO I love it when some of you people compare a movie to reality. Of course it's not going to work out like in real life but stranger things have happened in real life. You could ask these kinds of questions about almost every movie ever made.

Sarah Waters is the alias she created for herself in that town. Anyone who knows her there knows her as Sarah Waters. She has an apartment and a job, so presumably, the rental agreement says Sarah Waters. Her employment record says Sarah Waters. Ben knows her as Sarah Waters. She may even have an official form of identification that says she's Sarah Waters. The only person who knows her as anything other than Sarah Waters is her mother, and she's not in the area. When the police come, some of whom probably knew her, all she had to say was, "He must have mistaken me for someone else. He kept calling me Laura."

There would have been two ways in the pre-Internet world of the film for the police to have started wondering about Sarah's identity. Either they went to Martin's house and saw photos, or they looked into the investigation of Laura's disappearance, which may have included photos. Going to Martin's house would have been unlikely, as it's halfway across the country. And they have no real reason to look into Laura's disappearance. A cursory glance at who Martin was would have told them that his wife was named Laura and recently deceased. And if Sarah had told them that Martin kept calling her Laura during the home invasion, that may be enough for the police to say, "Well, the dude must have been a little unhinged and in his grief just thought that Ms. Waters was his dead wife."

Martin said she had called the police before. I assume His lawyers quashed the complaints.