There are a number of ways to send an SMS or give out a contact phone number without showing your personal phone number to the recipient. You can use a new number that's not associated with you, a temporary throwaway number or a text message sent from an email account. Even so, you may be identifiable based on the content of the message.
Anonymous Texting Numbers
Often the simplest way to send a text or make a call anonymously is to get a second number. That way, people won't know who is sending the message or making the call until you're associated with that number.
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A variety of services can do this for you. Google Voice, a free service, allows you to set up a free phone number for calling and texting, letting you access your texts and voicemails online or from a linked device or devices of your choice. You can also send texts and make calls online from your computer with Google Voice.
Another tool, called Burner, is an app that generates temporary phone numbers you can use for sending texts and making calls. Burner is useful when you want to give someone a number or send a text without revealing your identity, even if the recipient searches for the number you used.
A Device Just for Texting
In some cases, you may not want your texts associated with either your phone or computer. This is especially true if someone else has access to your devices.
In that case, you can send a text without your number showing or any notifications on your usual devices by getting a second device just for texting. You may want to purchase a cheap prepaid phone, like a discounted flip phone, for this purpose if you're only going to use it occasionally.
Remember that if you are physically near someone and they call or text you at your new number, they may hear your device buzz or ring if it's not silenced and realize you are the owner of the number.
Email to Text
Many cellphone providers allow you to send an email that is transformed into a text message. With this and an anonymous email address, you can send private text messages without needing a phone at all.
Of course, texts associated with your email address are only as anonymous as your email address itself. You may want to get a new email address from a free provider such as Microsoft or Google if this is your plan.