Eastern and Western are just two codifications, the total dialect variation within Armenian is similar to that within English and arguably much less than that within German.
As noted in the other answer, while the spoken dialect continuum developed over a thousand years, the minor fork in codifications only happened in the past two centuries.
One soft proof of their intelligibility is that you will hardly find a website which is in both Eastern and Western Armenian, nor will you find subtitles in television interviews.
Eastern vs Western
Here is the situation across the aspects that affect intelligibility:
Eastern Armenian is more conservative in terms of pronunciation. Western Armenian is more conservative in terms of grammar and in terms of orthography.
Both Eastern and Western Armenian have taken on new vocabulary for some concepts that emerged since their split, at least informally.
Soviet Armenian has taken on, via Russian, more internationalisms like desinfekcia, but Western Armenians will often understand those thanks to their better knowledge of French or English.
Western Armenian was more influential historically, while Eastern Armenian is much more influential since the Genocide.
How we know this
Armenian and Georgian have been written since the 4th century, so there is a reasonably sized corpus of "grabar", Classical Armenian from before and during the split, from which these assertions are derived. Even the pronunciation can be reconstructed, because of loanwords from Greek, Aramaic, Latin, Iranic, Kartvelian and so on, and using the comparative method.
Intelligibility in Practice
So educated Eastern Armenian speakers can read Western Armenian without any problem, because it is basically classic Armenian.
In theory, Western Armenians can understand spoken formal Eastern Armenian because the pronunciation is very clear, and they may be exposed to it.
But in practice there are many Russian words and other jargon in the street language that are difficult. This is more of a factor for intelligibility than any aspect of grammar or pronunciation.
But the use of Russian words as opposed to French or Arabic or English words actually has nothing to do with Eastern and Western.
Definition of Eastern and Western
Commonly people use "Eastern" to mean Soviet Yerevanian and "Western" to mean the diaspora, especially in the Levant.
But the Western/Eastern dichotomy is highly imperfect.
Firstly, it was always a dialect continuum, someone in Gyumri, someone im Kars and someone in Erzurum spoke similarly, no matter in which group we would place them today.
Secondly, some speakers in Russia, Soviet Armenia and especially Georgia speak with Western pronunciation, but because they were in the Soviet Union, they are familiar with the so-called Eastern orthography and vocabulary. Conversely, speakers in or from Iran, which includes those in places like India and Singapore, speak with Eastern pronunciation, but because they were not in the Soviet Union, they are not familiar with the so-called Eastern orthography and vocabulary.
So there are actually 4 basic groups: Western and not Soviet, Western but Soviet, Eastern and Soviet, and Eastern but not Soviet.
Forces of Harmonisation
Then there are those who moved across those groups more recently. To ask if Vartan Oskanian or Levon Ter-Petrosyan or their parents speak Eastern or Western Armenian is like asking if Elon Musk speaks British or American English.
In essence, a series of events - the Genocide, and today the flight of hundreds of thousands of Armenians from Syria and Iraq, whose families had found refuge there during the Genocide - has severely endangered Western Armenian, by killing most of the speakers and scattering the surviving ones across the world. It is strongest in Lebanon and Syria.
Those few Western Armenian speakers who do re-settle in the slice of historic Armenia where Armenian is still spoken inevitably learn Eastern Armenian instead, but also contribute words amd expressions and effectively force some dialect levelling. In the 1910s and 1920s a few hundred thousand refugees from the West arrived, roughly half the people in Armenia are descended from one, and that proportion is surely much higher in Yerevan. In Georgia too this is a non-trivial phenomenon, most Armenians in the Armenian-majority areas had fled from near Erzurum around the time of the Hamidian massacres.
Then there is net economic emigration, so for example an estimated 100,000 Eastern Armenian speakers now live in Istanbul, perhaps a similar number in Western Europe, and more in the United States, generally always to places where there was an earlier and continuing wave of Western Armenian speakers. Similarly the Armenians of Iran, while safe and very successful there, also emigrate in high numbers.
The net effect of this is that the Eastern and Western speakers keep coming into contact with each other, if not in historic Armenia then in places like Rostov, Boston, Los Angeles, Istanbul, Bulgaria, the Emirates, Zurich, San Francisco and so on. (Concrete examples where I know of speakers of both.) And online, and in music, film and television.
This contact, combined with a more than thousand years tradition of writing and emphasis on literacy arguably precisely as a means of bridging the imposed divide, has led to the constant harmonisation of the formal and informal standards, so that even new scientific words and so on are mostly the same in both variants.
Relative to Other Languages
This is in fact not so unique, half the world's major languages are polycentric. Standard formal British English and standard formal American English are quite similar, whether written or spoken. However, within each, the most unintelligible dialects are barely intelligible to someone who has learnt only the standard, let alone to someone who has learnt only another standard. We could say similar for German or Spanish.
And we can make an analogy regarding the fact that often one variant is perceived as archaic, and one as corrupted, when in fact the other variant is just as often archaic or corrupted, but in any case perceiving something as archaic or corrupted is also a tacit admission of intelligibility.
As an aside, one could argue that the divide has been the key to maintaining the language, as it is the only common language between Armenians living for centuries in the different spheres, very very few of whom are monolingual in Armenian. In that there are some parallels to Hebrew in the 19th century, and Romanes and Kurdish today. These languages were never very useful for, say, marketing, they are useful as a lingua franca between the different nodes of the community.
So, in summary, they are as mutually intelligible as the variants of the language we are writing now, but it is not so much a question of some metrics of the phonetic, grammatical or lexical differences, but of the sociopolitical reality, which is that the two standards do not exist in total isolation from one another, in contrast to, say, Homshetsi.