Private 5G networks has become a popular choice of various vertical industries to build dedicated and secure wireless networks in industry environments to deploy their services with enhanced service flexibility and device connectivity to foster industry digitalization. This article proposes multiple multi-domain solutions to deploy private 5G networks for vertical industries across their local premises and interconnecting them with the public networks. Such scenarios open up a new market segment for various stakeholders, and break the current operators' business and service provisioning models. This, in turn, demands new interactions among the different stakeholders across their administrative domains. To this aim, three distinct levels of multi-domain solutions for deploying vertical's 5G private networks are proposed in this work, which can support interactions at different layers among various stakeholders, allowing for different levels of service exposure and control. Building on a set of industry verticals (comprising Industry 4.0, Transportation and Energy), different deployment models are analyzed and the proposed multidomain solutions are applied. These solutions are implemented and validated through two proof-of-concept prototypes integrating a 5G private network platform (5Growth platform) with public ones. These solutions are being implemented in three vertical pilots conducted with real industry verticals. The obtained results demonstrated the feasibility of the proposed multi-domain solutions applied at the three layers of the system enabling various levels of interactions among the different stakeholders. The achieved end-to-end service instantiation time across multiple domains is in the range of minutes, where the delay impact caused by the resultant multi-domain interactions is considerably low. The proposed multi-domain approaches offer generic solutions and standard interfaces to support the different private network deployment models.