Can I pass my liability to a specific person upon death?
Say right now you approach any of your friend and tell them to take over your outstanding loans such as credit card debt or housing mortgage loan and to service the loan repayment. Chances are your friends will either politely reject you or laugh it off as a joke, essentially still rejecting your offer. But if you are offering to give them your investment account worth $10,000 with no strings attached, then it will be a totally different story now. Who will reject it?
It is the same concept here even upon death. You can "will" your asset away, but you cannot "will" a liability to any specific person. To put it simply, you cannot force someone to take on your liability after you die.
However, if you do "will" away an asset that has a secured loan, then when the beneficiary choose to inherit this asset, he will also inherit the loan together. For example, a beneficiary who accepts and inherits a property with an outstanding mortgage, he will also inherit the entire mortgage as well. As to whether the banks will extend the mortgage to this beneficiary is subjected to the bank's approval.
If a person dies with too much debt, he could die a bankrupt and the beneficiaries may receive nothing. This is because all outstanding taxes and debts have to be settled first (e.g. paid off) before the remaining estate can be distributed to the beneficiaries regardless of whether there is a Will written or not.
This is also where insurance can come in to help (e.g. hospital plan to defray hospital bills, mortgage insurance to cover liabilities etc.)