Identify two countries that operate under a form of democratic socialism?

"democratic socialism" isn't a political science concept - that is, there is not a specific governmental type (or subtype) of that name. Note that "socialism" (with a small "s") refers to an economic model, not a political model, with "Socialism" referring to a complete political/economic system. Thus, "democratic socialism" actually describes a form of economic model, not a political/economic model, which is what I assume the question is looking for.

If what you are looking for is a Democracy which follows socialism as an economic model, well, then, you're pretty much out of luck; there are generally but a very, very few countries which use socialism (as defined by political scientists and economists) as their economic model, and none of these countries could be called a Democracy.

Instead, what I suspect you are looking for is the following:

• A country which uses some form of Representative Democracy
• One whose foundational economic model is capitalism
• But who uses regulation and government policies to implement a large portion of the Socialist societal goals.

This kind of country is what United States mass media (and political discussions) generally label a "Socialist" country, despite this being a horrible misuse of the Word (and meaning) of Socialist.

Frankly, by the distorted US meaning of Socialist, the vast majority of 1st World countries follow "Democratic Socialism", including virtually all of Europe (save perhaps the U.K.), Japan, Korea, Taiwan, and even places like Brazil. Not to be snide, but saying "democratic socialism" during a US political debate is basically inferring that some country is almost a Communist country - that is, it's used as a slur rather than something which has a distinct meaning.

Objectively speaking, if you are looking for a democracy which has implemented a large section of the socialist-style societal goals, then the Scandinavian countries (Norway, Sweden, and Finland) are good examples, as would Spain, France, and possibly Portugal.