A visa to visit the island that switches countries every six months?

According to a New York Times article published in 2012, "you are not allowed to visit Pheasant Island", regardless of which or how many visas you have.

As both Spain and France are in the Schengen zone and (almost) all visas for tourists cover both countries, you will be covered by the same visa for the island which ever country is in power.

Working on the island however... you will need to find out which country deals with the visa for that.

Since the only permitted visits to the island are organised trips, you would need a visa for the country where your trip to the island began, which would also be a visa for the country administering the country at that time.

For example, as part of the French 2017 European Heritage Days programme, Hendaye commune organised a visit to Pheasant Island / L'île des Faisans on Sunday 17th September, which falls within the period of the governance of France.

At the time of writing, the 2018 dates for European Heritage Days in France have been announced (15th and 16th of September), but I cannot yet find a programme of activities. One non-official website has illustrated its page about the event with a photo of Pheasant Island which may indicate that there will be another visit this year.

Should the event go ahead, you could contact the organiser of this event to ask what the visa requirements are, but I would expect them to be a local official who (1) may not speak English well (2) has no specialist knowledge of the legal effects of condominiums on visa status. The visits appear to be aimed primarily at local people with an interest in the history of their town.

To enter in Spain or France, one needs the proper authorization. But the question has much to do with the "idealized" world where there are "(virtual) walls" in boundaries, and custom and immigration control at every entry point, and exactly at boundary point.

This is false. There are several kinds of boundaries, and often not very defined. The most known it is the territorial/geographic boundary. This could be no so well fixed. As example, Between Switzerland and France, and between Switzerland and Germany, some boundaries are "in the middle" of a rivers, and these dynamically changes according how much water flow in such rivers. Or from time to time Switzerland and Italy ratify new boundary lines: the official boundary is on drainage divide, but according glacier melting, this change other time.

The case in this question is similar: the territorial boundary change over time.

Custom have own boundaries. Sometime some villages are outside own country custom zone. This happens mainly on remote villages/islands. But there are also free-trade zone (mainly on commercial entry points). Not so relevant in this case (but if you put some alcohol or trade merchandise on the island, it would be interesting to see how the custom expect that you pay/not pay import taxes.

For this question, the relevant boundary is about "immigration"/police boundary. This boundary is not very well defined, and it is linked with local traditions (which were ratified by countries). This is often given for enclaves and river navigation on boundary waters, but also access to some paths and pastures for specific local communities.

I see no difference as you were hiking between France and Spain on Pyrenees: it is your duty to go to custom office and police on the first village you encounter, if you need to (entry stamps, taxes). The same, in this island: you eventually need to report to relative authorities. Being on the island, is like crossing a bridge (on boundaries) or following a road which get you to police/immigration office: you doesn't need a visa: when you got in the office, you can ask information and check if you can enter. In the contrary case, you should leave the country from the same route. Note: that zone is no-way a "nobody zone". You are not allowed to stay there. You can be forced to go to immigration office or to return back.

[If you want to camp in such island, or in any boundary zone, just notifying the local police station [in this case, of both countries] will really help you, so that you will not have the police control during night (they will already know if you are a "legal immigrant").]