What are financial prospects for a postdoc as compared to a PhD student without funding?
So is my financial situation going to depend on if I can secure a
  grant for my research as a postdoc or I should expect to receive a
  salary?


The question is not getting a grant or getting paid, it is getting paid by the institution you work for or getting your salary from a grant giving institution.


  How many postdocs do get paid?


(As far as I know) All of them. Depending on the country you are talking about it may even be illegal to employ someone without pay.
However, (as @virmaior pointed out) there are some "postdocs" in Japan that are not paid and have no work requirements. At least in Europe and the US this is not common and in most cases this is probably a bad idea for someone looking for a regular postdoc position (see my comment below).


  How likely is the success in either ways?


That depends on your field, the quality of your work, ... . We can not answer that but you can talk to your peers /supervisor about the job market to get an idea.


  [as a postdoc] I do not have to pay academic fees or any other fees to
  a university


Right.

One general remark: If you are not not getting a (paid) postdoc position or a grant you should try to find a job in the industry anyway. Finding a more senior position is usually much harder than finding a postdoc position.