This is an interesting question.
DonDiego was supposing that the solution proposed, f'rinstance in New York City, . . .
"Use the bathroom or locker room most consistent with their gender identity and/or expression without being required to show 'proof' of gender"
. . . would mean that those who were something other than male or female would simply use the restroom they preferred that day, a men's room or a ladies' room.
But, again using NYC as an example, the Big Apple recognizes 31 different "genders", at least ! [A NYC spokesperson stated that the list was not necessarily exhaustive.]
BI-GENDERED
CROSS-DRESSER
DRAG KING
DRAG QUEEN
FEMME QUEEN
FEMALE-TO-MALE
FTM
GENDER BENDER
GENDERQUEER
MALE-TO-FEMALE
MTF
NON-OP
HIJRA
PANGENDER
TRANSEXUAL/TRANSSEXUAL
TRANS PERSON
WOMAN
MAN
BUTCH
TWO-SPIRIT
TRANS
AGENDER
THIRD SEX
GENDER FLUID
NON-BINARY TRANSGENDER
ANDROGYNE
GENDER GIFTED
GENDER BLENDER
FEMME
PERSON OF TRANSGENDER EXPERIENCE
ANDROGYNOUS
Ref: NYC Government
It seems to simple old DonDiego that it all just means anyone may call oneself whatever one chooses, and request others apply one's preferred pronoun when informed of it, and use whatever restroom one chooses. But it might mean that some one or more of those 31-and-growing "genders" demand a separate restroom for themselves.
n.bFor the record poor old DonDiego
i. hasn't been in a locker room in decades and
ii. usually uses a stall when in public restroom
. . . so he's not generally too bothered by all this nonsense personally.
But apparently there's more to it than that.
"The NYCHRL [New York City Human Rights Law] requires employers[, landlords, and all businesses and professionals] to use an [employee’s, tenant’s, customer’s, or client’s] preferred name, pronoun and title (e.g., Ms./Mrs.) regardless of the individual’s sex assigned at birth, anatomy, gender, medical history, appearance, or the sex indicated on the individual’s identification.
Most individuals and many transgender people use female or male pronouns and titles. Some transgender and gender non-conforming people prefer to use pronouns other than he/him/his or she/her/hers, such as they/them/theirs or ze/hir. [Footnote: Ze and hir are popular gender-free pronouns preferred by some transgender and/or gender non-conforming individuals.]
This is the government as sovereign, threatening 'civil penalties up to $125,000 for violations, and up to $250,000 for violations that are the result of willful, wanton, or malicious conduct' if people don’t speak the way the government tells them to speak."
Ref: The Washington Post
Here's an example of sorta normal personal pronouns including some preferred by other than simple old-fashioned males and females.

However, The Washington Post article suggests the NYC Law may go lots farther. By making it illegal to employ a pronoun other than that which the object of that pronoun chooses, . . . subject to fines or civil actions, . . . and not limiting the choices at all, . . . many windows of opportunity open to those in a mood to sue, . . . like someone who chooses to be called "thee" because he's a Quaker, . . . . or someone who wants to be called "Milord" just because he prefers it, . . . . or someone who wants to be called "glugga", just because he can.
DonDiego plans on staying out of New York City, . . . just in case.