Although an ASP will generally manage all systems related tasks, loss of control and management oversight is assumed. You are trusting the ASP to follow good systems procedures for disaster recovery, user support, system upgrades, bug resolution, version control, and data security. Network dependence is another tradeoff. This raises concerns about network reliability, privacy, uptime, and connection speeds. Lastly, not owning the application could indeed be a tradeoff for some firms.
Ownership provides control, and control can provide comfort.
By definition then, application hosting implies some loss of control. However, as the application hosting market matures, confidence and comfort levels will follow. In the meantime, ask your ASP vendor to clearly document how they intend to address your concerns. In short, look for service level agreements, documented disaster recovery procedures, data security statements and descriptions, scalability benchmarks and proven technical support. Just as important, ensure the ASP you are considering is financially fit so that your data will not be held hostage in the event of a bankruptcy or other catastrophic failure.