Sovereignty is the word that comes up most in this debate, and it deserves a concrete answer rather than an abstract one.
If sovereignty means complete isolation from external influence on legislation, we don't have it now and haven't had it for decades. Through the EEA Agreement, we adopt about 70% of the EU's internal market regulatory framework. We're supposed to have "influence on the shaping" of this legislation but have no voting rights. In practice, our officials sit on committees and present viewpoints, but when it comes to a vote, they leave the room. The legislation is then adopted and we implement it.
Inside the EU, we'd gain full participation in decision-making. We'd get voting rights in the Council of Ministers, members of the European Parliament, and a commissioner in the European Commission. Small states have considerable weight in the EU system: in the Council of Ministers, where most important decisions are made, Luxembourg (population similar to ours) has real influence on outcomes.
This isn't a simple question with a simple answer. It's true that EU membership means some decisions are made collectively rather than unilaterally. But our current situation isn't independence — it's following others' decisions without full participation. Whether EU membership means more or less sovereignty depends on what you value more: voting rights on rules that apply to you, or the formal right to refuse — a right we exercise extremely rarely in practice.
Sources: The EEA Agreement, Art. 99–101; Treaty on European Union, Art. 16 (Council) and Art. 14 (Parliament); Althingi, "EEA and sovereignty" analysis 2023.