Money in politics is a hotly debated issue in almost every democratic regime. The debate, however, between public vs. private funded political campaigns is not settled. Most countries actually subsidize campaigns somewhat in line with their degree of state intervention in the economy. But I am not aware of a systematic cross country analysis of this issue. And for the Mexican case, most pundits immediately assume that public funding per se triumphs over private contributions.
My current take is that large fixed costs may justify subsidizing campaigns at the early stages of a democratic transition--although explaining why a regime would find that beneficial is also puzzling. However, subsidizing campaigns creates undesirable consequences and distorts the incentives of would-be politicians, political parties, etc. On the other hand, having only privately funded campaigns also has its problems, as it may over-represent wealthy special interests.
The argument for full public funding (that private money is "obscure") is odd: Following the same logic, the government should own most enterprises to get revenues and finance public spending. But most people would agree that it is more efficient to let private parties run their businesses and then just tax them.
Assume a country spends 1 billion in political party subsidies. Is this the best use of public money? How about allowing private contributions and then spending, say half a billion in monitoring contributions and campaign spending, or in susbsidizing only certain campaigns? An intermediate and interesting proposal would be: spend more on monitoring money flows, and only use public money to "match" the fund raising of candidates.