This is the information from the New Zealand Embassy regarding bilateral agreements:
"Visa-free access for New Zealand passport holders intending to stay for more than 3 months in the Schengen area is complicated.
Schengen area countries permit most holders of passports from outside the Schengen area to stay visa-free in theSchengen area as a whole, regardless of the particular country or countries, for no more than 3 months out of a 6-month period. (This is sometimes expressed as 90 days out of a 180 day period.)
However, New Zealand has bilateral visa waiver agreements with many of the individual countries in the Schengen area. These visa waiver agreements allow New Zealanders to spend up to three months in the relevant country, without reference to time spent in other Schengen area countries. The European Commission has confirmed that these agreements continue to be valid. These agreements thus effectively override the Schengen area restriction (which would otherwise be imposed on New Zealand passport holders) of no more than 3 months out of a 6-month period in the Schengen area as a whole.
The countries with which New Zealand has bilateral visa waiver agreements are:
Austria, Belgium, Denmark, Finland, France, Germany, Greece, Iceland, Italy, Luxembourg, The Netherlands, Norway, Portugal, Spain, Sweden
You can, therefore, move visa-free among the above countries for periods of up to 3 months in each country. If, however, you move to other countries in the Schengen area, the restriction of no more than 3 months out of a 6-month period in the Schengen area as a whole applies. If you have already spent 3 months in one or more of the above countries, your presence in a Schengen area country with which New Zealand does not have a bilateral visa waiver agreement may be challenged by local border control, police or other authorities. You may also be accused of being an overstayer when you leave the Schengen area, or when you enter another country outside the Schengen area."
So it would appear that for Kiwis the same applies in Spain as indicated above in Denmark.
"Despite confirmation from the European Commission that the visa waiver agreements continue to be valid, border and immigration officials in Schengen area countries are occasionally unaware of this and question New Zealanders’ rights to stay visa-free in the Schengen area for longer than 3 months.
Before you embark on your travel, you should contact the Embassy or Consulate of the Schengen area countries you plan to visit (not the New Zealand Embassy once you are overseas) to get the latest update on visitor visa requirements if you plan to stay in the whole Schengen area for more than 3 months."
I do not propose to check every country but if you do, please post the information here.