Tristam put together a great list here. The last point is the most important one. Never send money until you have verified that they are in fact a real landlord or property manager. This can be done by meeting them in person or by doing a video call and google searching them. If you are not able to find a linkedin profile or facebook profile or anything about them on the internet and they will not meet you in person or via google hangout or skype then it is most definitely a scam. Flag the listing and move on with your property search.
Regarding sharing your private information there are sites today that allow tenants to share their pertinent renter information without revealing sensitive information. Sites like www.tenantopia.com allow tenants to fill out a rental resume that includes all the information that landlords use to make a tenant selection decision without including your birthdate, social security number, address, or any account numbers. When a tenant inquires for one of my properties and provides me with a link to their tenantopia rental resume it makes my decision much easier and allows me to speed up my verification process and have the tenant moved in almost immediately after they view the property. Another great part is that I never have to have any of the tenant's private information in my possession.