Double extortion ransomware attacks are a form of cyber attack where the victims files are both encrypted and exfiltrated for extortion purposes. There is empirical evidence that double extortion leads to an increased willingness to pay a ransom, and higher ransoms, compared to encryption-only attacks. In this paper we model two important sources of assymetric information between victim and attacker: (a) Victims are typically uncertain whether data is exfiltrated, due to for example misconfigured monitoring systems. (b) It is hard for attackers to estimate the value of compromised files. We use game theory to analyse the payoff consequences of such private information. Specifically, we analyse a signaling game with double-sided information asymmetry: (1) attackers know whether data is exfiltrated and victims do not, and (2) victims know the value of data if it is exfiltrated, but the attackers do not. Our analysis indicates that private information substantially lowers the payoff of attackers. In interpretation, this suggests that private information is valuable to victims and a means to reduce incentives for criminals to pursue ransomware.