The main duty free area is on the second floor of the Departures area and is after security, so naturally only passengers can use it.
There is a second, smaller “Express” duty free store after Baggage Reclaim in Arrivals (on Floor 0), so you can stock up on supplies at the beginning of your trip.
The main duty free is open in the high season from 7am to 24.00, while in the low season it closes a little earlier at 22.45. The “Express” duty free store is open from 04:00 to 24.00 during the summer and 04.00 to 22.00 during winter.
The main duty free store in Alicante Airport follows the common model of forcing Departures passengers through a large area (at Alicante it’s over 8,000 square metres) with a total of 20 shops to the left and right. At first you will see make up and perfume/after shaves as well as clothing. And perhaps in nod to the popularity of tobacco with British travellers, there is a large tobacco section. There is an area for food alcohol, which has a “Thinking Alicante” dedicated area which gives prominence to local wines, spirits and food products.
When it comes to fashion, there’s the “Your Fashion Store” with the usual international brands as well as an enormous Desigual store. There are two Corte Inglès branded stores too.
Remember that “Duty Free” is a misleading term that carries over from the past. For European travellers, there is no additional discount off the prices shown. However travellers from outside the EU can get duty removed on some items and would need to show their passports and boarding passes to get that. For that reason, bear in mind that you may well find better value in normal shops at Alicante. However, Duty Free can be a lifesaver for convenience or for last minute gifts when returning home!
If you are resident in an EU country, then there are, in theory limits that apply to how much tobacco or alcohol you can bring back into the EU, under the concept that what you buy should be for personal consumption. However, practically speaking it would be nearly impossible to reach those limits (e.g. over 120 bottles of wine!!). Be careful though with spirits and tobacco as carrying a large amount could lead to attention from UK Customs if you are returning there.