In this paper, we consider the scenario in which mobile network operators (MNOs) share network infrastructure for operating 5G new radio (NR) services in unlicensed bands, whereby they reduce their deployment cost and extend their service coverage. Conserving privacy of MNOs' users, maintaining fairness with coexisting technologies such as Wi-Fi, and reducing communication overhead between MNOs are among top challenges limiting the feasibility and success of this sharing paradigm. To resolve above issues, we present MatchMaker, a novel framework for joint network infrastructure and unlicensed spectrum sharing among MNOs. MatchMaker extends the 3GPP's infrastructure sharing architecture, originally introduced for licensed bands, to have privacy-conserving protocols for managing the shared infrastructure. We also propose a novel privacy-conserving algorithm for channel assignment among MNOs. Although achieving an optimal channel assignment for MNOs over unlicensed bands dictates having global knowledge about MNOs' network conditions and their interference zones, our channel assignment algorithm does not require such global knowledge and maximizes the cross-technology fairness for the coexisting systems. We let the manager, controlling the shared infrastructure, estimate potential interference among MNOs and Wi-Fi systems by asking MNOs to propose their preferred channel assignment and monitoring their average contention delay overtime. The manager only accepts/rejects MNOs' proposals and builds contention graph between all colocated devices. Our results show that MatchMaker achieves fairness up to 90% of the optimal alpha-fairness-based channel assignment while still preserving MNOs' privacy.