Did she say why she needs the 'personal information' such as to give it to the child support office or what? As to your rights, you can ask for a court ordered paternity test, of course you may have to pay for it.
The good news is that genetic testing is much faster, cheaper, and can be performed far earlier than paternity testing could be done fifty years ago. It can be performed immediately after birth; in some cases, if the mother has had certain other testing done on her fetus, it may be possible to complete genetic sreening even before birth.
The bad news is that it's a whole lot more certain than it was fifty years ago. If it's a postitive test, unless you have a gentically identical twin brother(s), it's not unusual to get back a test result which states that the probability that you're the father is as high as 99.9995 percent.
If that happens, unless Mom wants to place the child for adoption, you'll need to start having a discussion about some of your "personal" information, because the next issue witll be child support.