The answer is no one, everyone, trust yourself and all of the above. I am almost certain that no algorithms are needed to do "proof of _" (just look at free software) emergent human behaviour can be codified gradually. Currently I think this is basically a UX problem - we need a way to pay the people who maintain the digital commons.
The game of printing more currency can be decided by a balance of the internal and external factors w.r.t. version control (where you are trying to decide relative value of contributions). You negotiate the best situation for yourself, the problem is symmetric and disputes cost too much attention so people lose interest and this thread of the project goes off-track so the exchange rates get worse or the supply of money dries up with no new versions... Basically, truthful stuff will probably win if we are careful about how we organize information since most people are quite pragmatic about stuff they aren't that interested in and money as a social construct only has meaning if we can reach approximate consensus on its distribution. This external pressure is what gets gradually explored and codified into (semi-)automated strategies for what is essentially investing.
The key idea though is that in a p2p system no one has oversight so everyone is in a situation of not knowing what to trust, my argument is that these social constructs are emergent because humans use them to communicate (to exchange assessments and make decisions on how to behave in the "real world"). Things that are taken seriously are considered legitimate and therefore the currency associated with producing that thing should be worth something but exactly how much is only decided when you find someone to trade with. One motivation for trade is if you depend on something, then you can buy insurance by lending legitimacy to the version you depend on (investing in that stock in some sense), that increases the legitimacy of the space of versions which means there is more to gain from making legitimate improvements to it - since there is someone who values them. Finding a bug in the most legitimate version means that you can print money at a very good exchange rate.
Such a system can be terribly inconsistent and still work good enough to be worth more than cryptocurrencies "secured" with various consensus mechanisms. Consistency is actually not incredibly important in the economic system I think. The more important thing is to avoid people prioritizing the measurement game over the reality and if the rules of the measurement game are too well defined then that is hard to avoid. Of course there is some balance here, I guess I am playing devils advocate by leaning a bit hard into the inconsistent economics. Probably some people would choose to maintain absolute consistency in order to obtain legitimacy.
This is becoming a ramble but was ment to be a compact braindump of what I am working towards with http://datalisp.is.