Either is a possibility.
Property-tax bills are sent by the municipality to a designated "representative" (koteishisanzei kyōyū daihyōsha 固定資産税共有代表者 ), who is responsible for making the entire payment and, if joint ownership is involved, ensuring that proper reimbursement is received. If one of the property owners has registered with the municipality as the representative, the bill will be sent there. Otherwise, each municipality is free to decide an order of precedence for designating a property-tax representative. Some choose to send the bill to the person with the highest proportion of ownership; others choose to send it to the person whose name is listed first on the register or to the owner with a local domicile. Multiple names may or may not appear on the bill (corporate ownership is also possible, of course). These days, the order of precedence followed by the municipality can often be found online. Only the representative gets the bill, but a few municipalities (Toyama City, for instance) may send confirmation notices to the other owners as well.
The owners always bear collective responsibility for paying the tax, however, so if the designated representative fails to pay, the other owners will eventually be dunned, probably with an added penalty. This principle of collective responsibility can lead to complications when there are multiple owners or when divorce or inheritance is involved (life events do not eliminate joint responsibility for paying property taxes).
