My landline works but the hand held phone stops working. I'm told I need a new phone. I found a set on Amazon for under $60.00.

This may sound like a silly question, but I really don't know. When I throw away
My old phone can people get information from it. I use it with my bank and it
Includes very private information.

Thank you.

Highly unlikely although it does depend on the make and model.
Landline phones normally only store details of numbers and nothing else.

As you say it is non-functioning then why don't you use a hammer on it to ensure any details are destroyed..

Also $60 for a simple landline phone seems high, I found several phones at BestBuy.com for between $20-$30
landline phones - BestBuy

But I agree with CurlyBen. Phones, even cordless p\ones only store phone number and records of when and how long the calls were.