Physical ChangesThe most obvious physical change experienced by a new Monster is the sudden growth of a large tail. There are a lot of other things that the sudden acquisition of a Tail can change in one's life. In addition, there are a lot of other effects that can occur because Tails often make changes to their host's body in order to ensure their mutual survival.
Major ChangesFirst of all, the host will have a large, sapient tail permanently attached to their spine. It is impossible for the host to remove the Tail through any means. In addition, the tail will be able to eat a certain diet and will change the host's digestive system and physical form if necessary to aid in obtaining food for that diet. Tails are somehow able to hold themselves up even if there's no way the host should be able to hold that much weight, but they will still feel some of the Tail's weight, which can be disconcerting. Due to the large size of most Tails, it would likely be necessary to either buy new clothes or alter old ones so that it wouldn't be restricted in its movement. Acquiring a Tail may also cause the host to grow other appendages or have their current ones altered. For instance, a tail that mostly eats sea creatures may cause the host to grow webbing between their fingers and toes. The traits of having tails are also passed from parent to offspring, and the offspring must each be given a special object so their tails don't become unstable.
Minor ChangesTails can cause a variety of other things about the host to change, from their hair or eye color, to their height. These are less life-altering, but still change their life.
Mental ChangesHaving a Tail changes a lot of things about the host mentally, too, even though there is generally no mental connection between the two. This is also done for survival.
Major ChangesSince damaging/destroying the Tail would be problematic for both parties, no Monster is actually able to do anything to attack, injure, or remove their Tail. Even if the Tail and monster hate each other, they can't actually physically damage each other much. The Tail can still bite the Monster painfully, though, so human Monsters often muzzle overly aggressive tails. Additionally, a Tail's special ability can also show up in the host, especially if their Core is near or within the Monster. This causes mental changes that depend on what the ability is.
Minor ChangesThere are likely other mental changes caused by the acquisition of a Tail, but there is little known about them.
Major ChangesHaving a sapient Tail can cause other things in the host's life to change, since they are a permanent part of their body. One of the biggest things is that the host has to figure out how to communicate and get along with a creature that is capable of complex reasoning without the ability for the two to talk in any way. Tails are usually only capable of animalistic noises, so this provides a significant barrier to communication. Sometimes the Tail doesn't like the host, so they have to figure something out so they can live.
Also, sitting on certain types of chairs might annoy the Tail because it hurts them. So the host may need to replace some of their furniture.