Eavesdropping (noun and adjective) is illustrated in the OED (entry not yet revised) with examples from 1601 to ante 1861. An 1853 quotation aptly says:
Eavesdropping consists in privily listening
a pretty good substitute for an explicit definition (which OED2 does not give).
This dual lexical item is attached to the entry for eavesdropper (noun). This word is given two definitions, which I will quote in reverse order.
The first definition is the familiar:
One who listens secretly to conversation.
The second is a cross-reference to a quotation of 1641 which gives a legal definition:
Evesdroppers are such as stand under wals or windowes..to heare news.
(The mention of ‘walls or windows’ is a clue to the origin of the term.) Eavesdropper goes back at least to 1487, when a certain Henry Rowley was adjudged by Nottingham juratores to be
communis evys-dropper et vagator in noctibus (a common eavesdropper and nightwalker)
There is of course a verb eavesdrop, but it is much later (1606) than eavesdropper; hence the latter is unlikely to be derived from it. Instead eavesdropper comes from a now obsolete noun, eavesdrop or (to modernize its Old English form) eavesdrip. This has nothing whatever to do with ‘privy listening’. As I said, ‘walls or windows’ gives a clue. It is the place where you stand to do your secret listening, or as the OED defines it:
the space of ground which is liable to receive the rain-water thrown off by the eaves of a building
the dripping of water from the eaves of a house.
The noun eavesdrop is very rarely recorded. The OED has an example from an Old English Kentish Charter and then no evidence till 1639 from the Wills and Inventories of Bury St Edmunds in a rather disguised form:
The..edifices..gardens, waies, water courses, easemts, ewse-dreepes, and appurtenances.
And after this two nineteenth century examples. The OED has a note in small type explaining the meaning further:
chiefly used with reference to the ancient custom or law which prohibited a proprietor from building at a less distance than two feet from the boundary of his land, lest he should injure his neighbour’s land by ‘eavesdrop’.
So while the completely transferred use of the verb, the agent noun, and the verbal noun are common and familiar, the original use is extremely rare (there are no examples on EEBO).
Into this gap of evidence comes an example of eavesdropping (noun) from the Essex Wills (1598 XII. 16):
a chase way or passage to the premises, with sufficient ground for the eavesdropping
This is both the literal meaning ‘the space of ground which is liable to receive the rain-water thrown off by the eaves of a building’ and earlier by three years than the first example in the OED.