Does buying domain privacy make the domain more secure against hijacking? Would it make the domain less secure against losing it, if the registar gets bankrupt? I am afraid that the domain could be seized to pay the debt of the registrar, if buying domain privacy makes the domain to be property of the registar.
Domain privacy will only hide your identity as a domain owner, in case someone do whois to your domain, they will see a domain privacy provider identity, rather than your real identity. You can use a fake identity instead, but it will give you some troubles when ICANN need to verify your identity.
Technically it may save you from some social engineering activities such as someone calling your phone number or send you an email (based on your domain identity) and trying to ask you something which considered important. Some people also can use your domain identity to make a scam website, mostly money games website.
Domain privacy may give you some extra security against your identity, but it will damage your reputation either, especially if you run a business website, people will think that it's a scam website because of those hidden identity.
Does buying domain privacy make the domain more secure against hijacking?
If you own the domain, you own it.
Would it make the domain less secure against losing it, if the registar gets bankrupt?
You should have records of your ownship besides the registrar
I am afraid that the domain could be seized to pay the debt of the registrar, if buying domain privacy makes the domain to be property of the registar.
In my opinion this is more of a legal question than a security question.
If you use domain privacy, there will be less public information about who you are and who owns the domain. If you make that information public on the site itself, it isn't much help. If the real owner of the domain is meant to be secret, there are a number of ways to use that information. It really depends on how public or private you mean for that information to be.
The owner contact on the domain reflected in the whois record for the domain is the owner of the domain.
However, some privacy services even shields the the owner contact. But ICANN simultaneously require an escrow database to be kept over all domain registration details. This information is uploaded to ICANN weekly and kept stored securely by the organization.
This information is used to transfer domains to other registrars in case the current one becomes de-accreditated, as in the case of bankruptcy.
The legal question remains however if the owner contact is indeed the "shield" and not the customer and someone challenges the ownership in a bankruptcy. IANAL.