The be- prefix in behead doesn't seem to match similar words like become, besmirch, or befuddle. Of course, the same prefix could serve different roles depending on the word. What role is be- serving here, and are there any other English words that use the prefix in this way?
We didn’t use de-head because we already had a verb behead by the time we started using de- to create verbs: behead was a verb in Old English, behéafdian.
So behead was already used long before the de- privative prefix came to be used productively in English. That didn’t happen until Modern English with a few productive examples in the 17ᵗʰ century but most coming from the 19ᵗʰ century or after. As Janus mentions in comments, the de- word meaning the same thing, decapitate, was imported in full with the de- already there, from Latin via French, in the 17ᵗʰ century.
There are many different possible senses of be- in verbs; the OED lists six different primary senses with subsenses. This here in behead is one of the rarer ones. Under be- sense 6c, it says that this privative sense of be- used to create behead is an ancient sense that means bereave of:
- Forming trans. verbs on substantives used in an instrumental relation; the primary idea being;
- a. To surround, cover, or bedaub with, as in becloud, to put clouds about, cover with clouds, bedew. Thence, by extension,
- b. To affect with in any way, as in benight, beguile, befriend. In both sets there is often an accompanying notion of ‘thoroughly, excessively,’ as in 2.
- c. An ancient application, no longer in living use, was to express the sense of ‘bereave of,’ as in behead, belimb, etc., q.v. Cf. 3, above.
Although 6a and 6b are still productive, 6c no longer is so in the living language. Another Old English verb formed using 6c was belandian, meaning to deprive of one’s land. However, this verb did not survive into Modern English.
Apropos de nada, the Spanish equivalent of the obsolete verb beland still very much exists in the verb desterrar, which combines the privative des‑ prefix with the noun tierra meaning land then puts that into an infinitive verb form. This is normally translated as “exile” in English, but sometime as “expel” or as a noun “expulsion”. There is also a substantive version, destierro, is famously found in “Cantar del destierro”, which is the title of the first canto from that most ancient of Castilian epic poems, El Cantar de Mio Cid.
Truly, el Cid was belanded of his lands by the King.
From wiktionary:be- (rare or no longer productive) Off, away, over, across
becut, bedeal, betake, bego, behead, belimb, beland, benim, bereave, besleeve, betrunk
From Saxon Be-=('to') and heawian=hew=cut off. The original is "Beheawon heafde" to cut off the head.
From "Beheawon heafde" to "Beheawonheafde" to "Beheafde" to "Behead". ( somewhere through the ages some lexicographer did away with the double 'hea' and further surgery carried out resulted to "Beheafde" and later "Behead".)
Be- [Germanic 'be'] is often used as a prefix. When prefixed to verbs, be- frequently expresses an active signification, as behabban to surround; begangan to perform or dispatch, &c. Sometimes be- prefixed indicates no perceptible variation in the sense ; as belifan to be remaining, or over and above, begyrdan to begird or gird, as in sprengan and bespren- gan to sprinkle, or besprinkle.
(1726 English Etymology Dictionary) (Anglo-Saxon Dictionary 1888)
This type of be- denoting separation ( only in certain older verbs) is in my view connected with German ab- (one meaning can be separation) and Latin ab-. Somehow the b has changed its position from after the vowel to before the vowel. This can be explained as metathesis or a right-left phenomenon.