It would seem that in connection with Purim, the obligation is one of "Mikra Megillah," "Reading the Megillah." The definition of communal "Reading" is "Mashmia L'Ozen," "Making known to the Ear;" that is, transferring information from the parchment where the living words lie dormant, to the (heart and brain via the) ear. The mechanism in Jewish Law which allows this to work is the principle of "shomea k'oneh," "the one who hears is as the one who speaks;" with the speaker causing the information to pass into the realm of "dibbur," "expression." Once it is in that realm, it is available to all, including himself; in a sense, at that point, all are equal.
However, in the case of Sefirat HaOmer, the nature of the obligation is to count. The Gemara in Menachot 65B, cites the verse which says "U'Sfartem lachem," "And you shall count for yourselves," implying that the counting should be done by each individual. This means to say that there is an obligation on each person to count, to give expression to his awareness that another day of his or her life has come, a new opportunity for personal spiritual growth.
It is somewhat analogous to an ice-cream cone, although the category of "Birchot haMitzvot," blessings for commandments, and "Birchot haNehenin," blessings on taking pleasure from the world by eating and drinking, are somewhat different. If I am holding that delicacy and am ready to enjoy it, another person cannot make the blessing which one makes on ice-cream cones for me… PAUSE for quick review of basic idea of blessings:
If I am planning to enjoy this delicious object, simple courtesy (and Jewish Law) require me to ask permission from its provider, the Creator of the Universe (rather than the manufacturer of the cone, who has presumably been paid), to do so. This is done by a preceding "bracha," or blessing; in this case, the "Shehakol nihiya bidevaro," "that everything came into existence by His word" blessing. And also, to say "Thank You" by means of a "following" blessing; in this case, the "boray nefashot," "the One Who created many souls" blessing.
The other fellow presumably also has much to be thankful for in his life, but not my ice-cream cone!
Similarly, in the Sefirat HaOmer context, in a sense, it is "my time," granted me by my Creator, to use to make myself a better person, that I am counting, and therefore someone else cannot count for me.