They are not required to tell you this. As you could imagine, such a requirement might result in notification to an accused and then someone being motivated to destroy evidence or intimidate (or eliminate) witnesses.
What I suggest you do is contact a bail bonds company to inquire into a bail bond being necessary. The bail bond company will then call over to the police department to check on the status of the investigation and the police may slip out information to the bail bond company that they would not directly tell you.
The opinions rendered herein are based on general principles of law. Laws vary from jurisdiction to jurisdiction and there are often numerous factors which can render advice or an opinion inapplicable. You should NOT make any decisions about the handling of a legal matter without first directly consulting with an attorney about the particulars of your case.
There is no requirement that the police notify you if there is a police report about you. If the District Attorney actually files criminal charges, then an arrest warrant would be issued (on a felony) and you could be arrested. The best thing to do is hire a local attorney - someone who can find out more information that you can on your own.
Short answer is that they don't need to tell you if they're continuing to investigate.
Whatever report they end up with, if they believe there is enough evidence, will then likely be sent to the District Attorney's office for review and possible filing of charges. If they decide to file charges, then you will be notified, but up until that point, they are not required to tell you whether they will be pursuing an investigation. Likely the best the police department would be able to tell you was whether any report was forward to the District Attorney.
It's frustrating, and uncertain, but as was noted, given that some people out there would use that information to potentially destroy evidence against them, police agencies are not likely to want to spread that sort of information around.
The advice provided in this answer should not be considered legal advice. There are many additional details to be discussed before a complete picture can be obtained and before advice can be given. Contacting an attorney for a confidential consultation should be considered.