A common objection I hear upon criticizing someone's claim to be of a certain sex is that people have the right to define themselves on their own terms, and to disrespect this is to disrespect the person.
In general, it is of course a good idea to acknowledge people's desire to be identified a certain way. A simple example would be, if someone asks to be referred to as Ricky, but you keep referring to them as Patrick (on intention), you're being obnoxious at best. You're being disrespectful, and possibly even doing it to bully them, or at least giving them the feeling of being bullied. Even if their name is officially Patrick, it makes no difference.
Another example might be choice of words used to refer to a person with a certain condition. A person who is overweight may prefer to identify themselves as "plus sized", or "overweight", instead of the usually rude-considered "fat". Or they may find such polite-considered synonyms to be disrespectful, as they could be said to carry the implication that the condition is something embarrassing, and therefore prefer to be boldly called "fat" or even a "fatty".
Similarly with people on the autism spectrum, who may prefer "person with autism", "autistic person", "aspie", "autist", or any other variation. (And if you know a bit about autistic people you may be aware that this can be very important to some of them.)
There is, however, an important difference between such situations, where a person merely prefers one of many synonymous terms, and situations in which a person claims to be something which, as far as the generally accepted definition of the word goes, they factually are not. For instance, I'm ethnically half German and half Turkish, and have citizenship of the corresponding countries. If I referred to myself as an Asian, I would either be stating something about myself that is factually inaccurate, since Turks are generally not understood as "Asian" (let alone Germans), or I would be implicitly using a definition of the word "Asian" that is different from what it's usually assumed to be, such as one including Turkish people.
If I demanded people to consider me an Asian at all times, despite my Turkish/German ethnicity and citizenship being well known, asking them to treat me the same way they treat other Asians, then I would not only be personally using an altered definition of the word "Asian", but demanding others to adopt this new definition into their vocabulary, replacing the old.
In short, this would not be merely about my self-declared identity anymore; it would cross over to demanding society to accept an altered definition of a well-known word, over which I have no de facto authority, and which is in fact shared by a lot of other people, i.e. actual Asians.
When a male-born transgender individual demands to be considered a woman, they are not just using their right to self-declared identity. They are demanding all of society to accept an altered definition of the word "woman".
Transgender activists should be open and honest about this.
After it has been made relatively clear what the new definition for the word "woman" is going to be, a rational discussion could be had on the merits of accepting or refusing this new definition. Until then, "transwomen are women" is only a shoddy mantra.