Why cant it work the same as a bank? The bank doesn't magically create money, it uses money from deposits to lend out. That could be done with bitcoin. Also, an individual very well can have the same level of claim as a bank to say...your house. You could sign a loan to an individual and put your house up as collateral. The issue is not whether that can be done, but actually collecting on a default. That process gets very expensive and requires lawyers, which may not be as easy for an individual to pursue.
Another issue with loans denominated in bitcoins is most likely, that person is cashing out the coins to purchase something. With a dollar that isn't a huge deal as deflation is a percent or two per year, but a bitcoin that can gain huge gains overnight, there is a very large risk that the borrower could never pay the loan back and that the collateral quickly becomes worth less than the loan amount.