Shares aren't traded directly. They are bought and sold through the same kind of "retail/wholesale" supply chain as TV sets or apples. Stockbrokers are the "retailers" and they simply take instructions from buyers and sellers and do a trade with the market maker ("wholesaler") who sets the spread (buy/sell price differential) for a share - typically by assessing the demand for trades he receives from different stockbrokers. Always, the buyer pays more (ie the top end of the spread) than the seller gets (bottom end of the spread).
Market makers may find buyers to match sellers or they may keep a float - ie they are buying shares without selling them, or vice versa. The float is necessary to ensure that sellers can always dispose of their shares even if there may be currently no demand for them - and it represents both a risk and a possible reward for the market maker.
So if you include market makers as 'buyers' then yes, every share transaction has a seller and a buyer, albeit at different prices because of the spread. But if you mean that every share sold by an owner is bought by a new owner, that isn't the case - at least, not in real time.
Because of the role of the market makers, a day's trading could theoretically be all sales and no buys, as PlusNet's almost was yesterday. A more typical day would see a greater variety in the number of sellers and buyers - see
this list of PlusNet trades today where the assumed sells are coloured red and the assumed buys are in blue. Note that a big seller of 73764 shares could only get 101p for them at 10.42am but it looks like the market maker found a buyer for them at 106p an hour and a half later. But for that period of time, such a large chunk of shares without an immediate buyer in sight represented a potential risk to the market maker, hence the low price he was willing to pay for them. For the time that the market maker held on to those shares, they were part of his float. It would of course be interesting to know who the seller was; and even more interesting to know who the buyer was. It's possible that the buyer would be an existing large institutional holder of PlusNet shares who feared that allowing such a large number of shares to remain in the float would drive down the value of his existing shareholding.
More information
here.
Simon