On desktop, the whole of Twitter's private messaging is done within a modal.
The initial page for DMs shows you your current threads and lets you start new ones with a "New Message" button.
A messaging thread here looks like you'd expect it to. You can start typing your message into a field, or use the action buttons to add Gifs or images.
There's also an emoji button for quickly adding popular emojis to your message.
Sometimes, you get default messages you can send to accounts. These default messages have titles and short descriptions.
For example in this case, there's one called "Culture," and the description is "Literary insight and cultural commentary."
Clicking on one of these options simply sends what looks like a normal direct message.