If you are going for a "social market economy" kind of society - e.g. a free market but elements of what would be called "socialist" in the US - then there is a place for huge businesses that are run privately (i.e. not state controlled). E.g. Wallmart, Ford, Microsoft...
I would say the government should step in, that is the public should gain control, if the enterprise becomes utility-like. If you "need" their services to participate in society at a reasonable level. Or if many people build their businesses on them. This feature of "being essential" is not neccessarily the same question as whether a company is a monopoly or not. The end 1990s Microsoft was maybe a monopoly, but it was not a utility or infrastructure.
OTOH, I'm not sure you could classify Google or Amazon legally as Monopolies, but they have created awesome (in both senses of the word) infrastructure. I don't think this infrastructure should be entirely in the hands of a company or a few people. We also don't let owners of toll roads make the rules of the road.
So I wouldn't neccessarily nationalize Walmart, FedEx, Ebay. But I would regulate large Telcos for example. The model here in Germany seems to work quite OK. The market is deregulated, most of the infrastructure is still owned by the former state companies (Deutsche Bahn, Telekom, ...), and these rent the resources to other companies. But there are regulatory bodies that ensure that e.g. the prices are fair. It's not perfect, but it seems the resulting situation is preferable for customers (having lived in the US and Germany, I'd say both public transport and telecommunication is on average cheaper and better in Germany, and I think it is a result of this model.)
Twitter is a possible target. It doesn't seem profitable as a business anyway. But if someone decided that we as a society needed "a twitter" (I don't know if we do...), then maybe it could continue in this way.
Except by your own reasoning you don't "need" Amazon.
> Twitter is a possible target.