What about conversational ability in the dead languages? Did it exist at all?

Yes, of course, if we're talking about dead languages that were once living spoken languages

How often would someone who knew a dead language get the chance to speak it with someone?

There are two different things you're dealing with here, and depending on which one you're looking at, you're going to get different answers. Of the two divisions, one is liturgical languages, which would generally be languages like Hebrew (until it was revived) and Sanskrit, as well as Qur'anic Arabic (separating it here from spoken varieties, even including Modern Standard Arabic). The other is written lingua francas. These would be things like (non-religious) Latin and Classical Chinese, the latter of which in its various forms was used to communicate between places like Japan, Korea, Vietnam, various Chinas and beyond.

Do we know how widespread that level of competence in those languages was?

The thing is, even though languages like Latin and Classical Chinese are "dead" as typically characterised, in the period in which they were in wide use for written records, communications and academic work, there was still a pretty high expectation of proficiency. For example someone who would be involved in any sort of work that would involve writing Classical Chinese would also likely have passed the civil service exams (or that nation's equivalent) which very much demanded a high degree of proficiency. Generally, you'd study the language and relevant texts for years before being in a position to produce lasting texts. In the case of Latin or Greek, study of the classics (and thus the languages) were a regular expectation for anyone seeking to be educated.

So in that sense, yes we do have records of the general number of people passing these sorts of exams, or records of what higher education has been in past periods which would thus require such study.

There seem to be a lot of texts written in certain languages in eras when those languages didn't have any native speakers and were only used for ritual or scholarly purposes

So, scholarly purposes I've addressed above; You learn Latin and/or Greek as part of your general higher education, and these are the languages you write in.

Hebrew is another matter, as would be Liturgical Latin or Biblical Greek. These fall under the "ritual purposes" category. Clergy (or the equivalent) or religious scholars would have training in the language/s, but the average person would not. This is ignoring minimal requirements such as Hebrew for the purposes of a bar/bat mitzvah. But this is going to me a bit more limited than the "scholarly purposes" side of things. In modern times, we still see this division; English tends to be the scholarly language in most international fora, while religious texts and the interpretation of said texts tends to be the realm of the religious elite. That hasn't changed much.

This is getting long so I'll stop now but let me know if you want clarification. It is a broad question as you mentioned, so I've tried to keep the answer similarly broad. Let me know if you want to get into more detail on any of it. Hopefully this helps you get a sense of where you want to go w/ the enquiry.