You should make an effort and define " in a decentralized way" and get rid of the nonsensical idea that mining would serve that purpose better than giving them away for any resource (a fundraiser achieves the same here, there is also a resource required to obtain coins (money as opposed to hashing power (hashing power = electricity/money + mining equipement = knowledge + money))).
Edit: I think you are talking about user issued assets / custom tokens on Counterparty... The same as above applies: Distribution via mining is not more decentralized than given them away for any other resource. Not idea whether you could let someone mine user issued assets but I also do not see a reason / use case.
And the Bitcoin blockchain is used with Counterparty for processing tx.
Thanks for your help, Delulo.
Yes, I am talking about user issued assets. I understand what you mean about mining being just as centralized of a way to hand out tokens as giving them in exchange for another resource (if not more centralized). But I think we are thinking about different methods to issue tokens for resources. Like mining, I was hoping there is some way to distribute tokens of a particular asset according to some set schedule and to any node, rather than one person creating all the tokens and giving them out according to those rules. The first case I would think of as decentralized distribution, and the second as centralized distribution.
Maybe I am just not sufficiently familiar with counterparty, but I thought only one entity could issue additional tokens for a particular asset at a time. Is this correct? I know you can transfer the rights to issue new tokens, but I'm thinking it would be interesting if assets could be created in a process similar to mining (although, as you said, energy doesn't have to be the resource consumed), where tokens are distributed in a certain schedule and not through a central entity.
As far as I know, the protocol dictates tokens are send to the address that originally requested the creation of them. It's then up to the issuer to decide to distribute them in whatever manner they see fit.
I think the closest thing in the protocol to some form of automatic token distribution is the burn message
https://github.com/CounterpartyXCP/Counterparty#burn used for the initial proof-of-burn. The only similarities to traditional proof-of-work distribution really were that 'difficulty' increased over time- less and less 'rewards' were generated and after the end-block was reached, the supply had been permanently capped. I don't think you can use the burn message for your own token though.
You always have turing complete smart contracts with SEND opcode for distributing tokens, (once live on mainnet)- You're free to play around with implementations on testnet now though. You could probably make something closer to the '1 cpu, 1 vote' idea than the version of bitcoin that currently exists today
Another basic idea could be generate a list of public keys derived from weak passphrases and send token(s) to each, in effect they would be mineable by bruteforce. You could add fun social/gamification elements into play, like revealing part of the input used to create the key (or the entire key itself for the first user to sweep the token) through counterparty broadcasts, public puzzles. twitter etc.