Do I need to send a check for the amount I'm amending when I send in my 1040x form?

Yes, and there are other options as well.

Line 19—Amount You Owe

You can pay by check, money order, credit or debit card or the EFTPS. 

To pay by check or money order: Send your signed Form 1040X with a check or money order for the full amount payable to “United States Treasury.” Do not send cash. Do not attach your payment to Form 1040X. Instead, enclose it in the envelope with your amended return. On your payment, put your name, address, daytime phone number, and SSN. If you are filing a joint Form 1040X, enter the SSN shown first on the return. Also, enter the tax year and type of return you are amending (for example, Amended 2015 Form 1040”). 

If you dont, will the IRS send you a letter for the amount you owe?

If you don't make arrangements to pay your 2015 tax liability by April 18, 2016, you will receive a bill by mail from the IRS and will begin to incur late fees and penalties.