I would slightly revise your premise. The relationship between individuals and other individuals is fundamentally different than that between individuals and government. In the former, mutual benefit, compromise, or coercion might be used to resolve conflict. In principle, government's job is to take coercion out of the picture to level the playing field. It achieves this by using another form of coercion (i.e. the force of law) but (theoretically) applies this with justice.
government can compel you to do things.
The concept of hierarchical communities arises naturally from our inability to negotiate terms with every person we come across. A group of people agree to terms and that binds their community. Communities then negotiate with each other and so on until we arrive at the level of national governments, regional organizations (EU), and eventually the UN.
I find Locke's concept of property rights fundamentally at odds with the idea of the right to life or property in general. Locke's view considers each of the intrinsic rights as solitary with no interaction. I see them as necessarily related: we invest part of our life (i.e. time) to earn property. Thus, if someone takes my property, they have infringed my right to life. The balance I choose between life and property is governed by my right to freedom. People are free to work themselves to the bone to earn property just as they are free to live a life of leisure and not earn anything; most choose something in the middle. Government or individuals with the ability to take my property then overthrow my rights to life, liberty, or property. Very few people would invest in an education if they did not have the right to keep the improved wages they expect to earn. Relatively little would get done if person A could earn more by busting his ass only to have person B come and take it. Locke merely proposed a theory - he did not arrive at the only logical conclusion. His conclusion is akin to a linear model which neglects interaction effects between the rights.
You are right, the relationship between people is different, it's a promise (or as Hobbes would describe, a covenant), while the relationship with government is a social compact. But it is interesting you find Locke's thoughts on life, liberty, and property at odds with those ideas because he pretty much invented it... and brought them together in a pretty spectacular fashion. In fact, he viewed property largely as a person's labor (which as Adam Smith showed pretty convincingly, all property is basically labor traded), and Locke addressed the interaction between life and labor. Because the reality is, they are all basically the same- a person's labor is his life, his life is his liberty, hist liberty is his labor, and so on. It's only really split for reasoning purposes.
Just remember you do not have an absolute right to life, liberty, and property... unless you live in a state of nature where no government exists. And if that's the case, what good is it if it can't be protected and appreciated? That's why the Founders designed a government that would maximize those natural rights through a legitimate constitutional order. Don't kid yourself, our system was designed not for you, but for everyone. Everyone gives up a little, the "right" to act on their absolute liberty, for the benefit of all. That's our contract in a nutshell. Each side gives a little for mutual gain, with a promise not to deviate from or abuse it. The bold is what we need to focus on, not some imaginary notion of unlimited liberty. The Constitution is supposed to be a restraint on government power in that contract, but that power needs to be continually debated and checked.