In the case where it's just one person's word against another, the police can charge a person with a crime. In many instances, it may be just that one person's word against the other. There may not be any evidence. Otherwise, for example, any injuries in cases involving domestic violence or documentary evidence, such as pictures or videos or text messages, emails, social media posts. Even if there's a smoking gun, there's often not a smoking gun. For that reason, many cases do come down to one person's word versus the other.
If that's the case, then the determination would be made based, in large part, on credibility in terms of who's believed to be more credible than the other. Fundamentally, yes, the police can charge a person with a crime, even if it's just one person's word against the other. Having an experienced Pennsylvania criminal defense attorney will be your best ally to help you understand and navigate the criminal process. They should be involved as early as possible if you're facing charges in Pennsylvania.