Who are the modern philosophers, and do we think historians will recognize them years from now?

Since r/askhistorians has a 20-year rule, I do not believe I can talk about present-day philosophers. But let me know if I am misinterpreting this rule. (As an academic employed in a philosophy department, I definitely have an answer to that question.)

Modern philosophy seems to be unheard of to anyone outside philosophy academics. Is that how it would be in Ancient Greece too?

Let's take Plato as a case study.

Was he well-known in ancient Athens? Absolutely. We have fragments from seventeen different comedies that mention Plato (usually unflatteringly or poking fun at him). Aristophon wrote an entire comedy called Plato. (I'd love to read it -- a shame it didn't survive...)

This was facilitated by a host of wider social circumstances. Firstly, every ancient school of philosophy was in competition with the others: they all vied for support, wanting to attract followers. Especially in the 4th century BC (i.e., the time of Isocrates, Plato, Aristotle, etc.), Athenian philosophers were in the business of providing "citizen training" to the sons of wealthy people. Plato depicts in the Alcibiades Socrates trying to persuade Alcibiades to follow him -- in order Alcibiades will be even better when he presents himself to the Athenian assembly.

Another factor was that philosophers were politically engaged. The social factor responsible for this was that ancient Athens, if we are going to continue to use Plato as a case study, put a lot of emphasis on active citizenship and partaking in the leadership of the city. We learn in Aristotle's Rhetoric that Plato did attack prominent politicians publicly: "Thus for instance, Eubulus, when attacking Chares in the law courts, made use of what Plato said against Archibius, namely, 'that the open confession of wickedness had increased in the city'." (We do not know who Archibius was or why Plato had been attacking him.) Plato himself famously went to Syracuse and had some unclear, much-debated relationship to a rebellion there.

When did the concept of philosophers as depicted in Ancient Greece go away, and why?

We're going to have to make some large generalizations in order to answer this. Some but not all factors are:

1. Philosophers no longer explicitly vied for students. There is no incentive for major philosophers such as Spinoza to do this. Spinoza was a lens-grinder by trade, who had been so turned off by the response to his Theologico-Political Treatise that he never even wanted to publish anything he wrote after that. He wore a signet ring that said caute (Latin for "cautiously"), embodying his preference for secrecy.

2. Philosophers continue to be socially or politically engaged but there is no special need or reason for them to be. Whereas someone like Plato simply had to be, because he was an Athenian citizen and active citizenship was expected of him and because he had to compete for more students, there is no need for Spinoza or Nietzsche to be. There's nothing stopping Descartes from working at the court of Queen Christina of Sweden. But there's nothing compelling him to do so.

3. As we leave antiquity, philosophy itself becomes less important. Central questions that matter to a human life are answered by theology, many times in concert with philosophy, but not always. After the medieval period, when it is possible for philosophers to work without also being theologians (e.g., Descartes), philosophers sometimes don't have anything to say about ethics or how to live a human life. Descartes' work is a good example of this. Why would an ordinary person pay attention to Descartes' work?

(This last point is an example of an extreme generalization being made for the sake of answering the question. We're overlooking the entire philosophical tradition in the medieval period in the Arabic world, where philosophers often disagreed with theologians, over issues such as the resurrection of the body, the eternity of the world, God's omniscience, etc.)

But philosophers aren't really unknown, even in the 20th century. Even late in the 20th century, see David Foster Wallace's novel The Broom of the System, which engages at great length (and often openly) with Wittgenstein, for instance. Sartre's work in Paris after World War 2 was so well-known and so popular that in an anecdote conveyed in Existentialism is a Humanism, he explains:

The essential charge laid against us is, of course, that of over-emphasis upon the evil side of human life. I have lately been told of a lady who, whenever she lets slip a vulgar expression in a moment of nervousness, excuses herself by exclaiming, “I believe I am becoming an existentialist.”

Our society has changed so much since the ancient world that it is now possible for philosophers and other academics to belong in an ivory tower, but in practice, I don't think that they are in an ivory tower.

Probably the best, most fruitful way to follow up on this is to specify more narrowly what trends we want to talk about. I do think that we're liable to drastically overstate the changes made to the role of the philosopher in culture and society. The trend over the course of history has been that philosophy is made more and more widely available to people, but the problem is that we're speaking about dozens of traditions over the course of thousands of years, and we're generalizing tremendously.

First off, let me note quickly that I’m not entirely sure exactly what sort of information you are after. I’ve jotted down some things that I think you might be interested in below, but please don’t hesitate to ask follow up questions if I’ve radically missed the mark. Let me also say that everything below generalizes in massive ways, and no doubt reflects my own training as an American philosopher in the analytic tradition.

The notion of philosophy as a career is alive and well, though it is a challenging one in various ways. Generally speaking, professional philosophers are employed as professors at a college and both teach and publish books or articles on their areas of expertise. One of the challenging aspects of the career is the highly competitive nature of both the job market and top level graduate school programs. As a rule, you need to get into a very good graduate school to have a good shot at an academic position in philosophy, and consequently graduate school admissions at such institutions are incredibly competitive. As an anecdote, I am currently enrolled as a PhD student in philosophy at a major American university. There are 5 students in my cohort, one of whom is likely leaving the profession. The year all of us were admitted, the university received in excess of 160 applications. So, there are probably at least a few philosophers employed at a university near you, and they went through a pretty strenuous process to be able to do and teach philosophy professionally. Each year, such philosophers are publishing books and journal articles, attending conferences, and generally just thinking about philosophical issues. While every philosopher will have a grounding in the canonical philosophers, the history of philosophy is a specialized subfield. In that subfield, we think not just about what the philosophers of the past said, but also what they might have to teach us about issues of contemporary philosophical interest. We are not, as a general rule, particularly interested in things like the biographies of past thinkers, except when that biographical information throws light on a philosophical point. As an example of the latter, it is often noted in work on Descartes that he was educated by Jesuits, because this helps us understand aspects of his philosophical work.

Philosophers today do see themselves as working in the tradition of Plato, Aristotle, and the rest. Sometimes, this identification is explicit. Positions which endorse the existence of transcendent entities are often called Platonist or Platonic. So, for example, a Platonist about mathematical ontology might hold that numbers are abstract but real entities which exist outside of space and time. Aristotle is even more commonly invoked. Philosophers of science who invoke essences or causal powers are sometimes described (or self-described) as neo-Aristotelian, and similar points could be made in the philosophy of mind and the philosophy of language. The most well-known variety of neo-Aristotelianism is probably the rise in interest in virtue ethics in recent decades, such as that of Rosalind Hursthouse and Martha Nussbaum. Shifting forward in time a bit, a great many contemporary philosophers are sympathetic to the work of David Hume, one of the canonical early modern European philosophers; David Lewis, probably the most influential metaphysician of the last 50 or so years, is a great example of this. And so on and so forth.

The consonance with the past is also not entirely a matter of identifying with the past. Philosophers today are still interested in many of the same issues as past thinkers. For example, the great political theorists of the last 50 or so years are folks like Robert Nozick and (especially) John Rawls. Nozick, for his part, begins his seminal Anarchy, State, and Utopia by laying out an avowedly Lockean conception of natural rights. Rawls, for his part, is deeply influenced by various thinkers in the liberal tradition, perhaps especially Kant but also social contractarian theories generally. More broadly, thinkers like Nozick and Rawls are interested in some of the same questions that animated Locke, Kant, and Plato. What is the purpose of the state? What is the ground of political obligation, if any? What is it for a society to be just? We could make similar points about every other major area of inquiry in philosophy.

I’ll also add quickly as an example of the continued influence of philosophy upon more practical pursuits, that Nozick has had a profound influence on the development of modern libertarianism, and that Rawls was awarded the national humanities medal by president Clinton in recognition of his work on political liberalism (in the broad sense in which every American political party is liberal). So, just occasionally, philosophy still has the power to capture and move those who are not necessarily interested in becoming professional philosophers.

I’ve ranted quite a bit here, and hopefully much of what I’ve said has been to the point. Let me conclude by noting that today, philosophers are often divided into two broad camps. The first is so-called continental philosophy. A paradigmatic thinker within this tradition is Martin Heideggar, and a paradigmatic interest is phenomenology. I don’t know much about this tradition, however, as it is not my own; I’ve never so much as glanced at anything written by Heideggar, despite being ABD in a philosophy PhD program. Most of my professors are in the same boat in this respect. This is because we are all analytic philosophers, meaning we are working in the tradition of thinkers like Frege, Russell, Moore, and Anscombe. We are all hyper focused on clear and precise argumentation, and many of us are preoccupied with the philosophy of language, logic, and what science has to teach us about philosophically important issues. This dividing line is not necessarily hard and fast, but it is commonly referenced and is worth mentioning.

If you want to learn more about specific periods in philosophical history, there is a history of philosophy section in the subreddits booklist.