How should private tutors file at the end of the year?

To report your self employment income you will fill out schedule C in your personal 1040 tax return and pay SE self employment Tax. You can enter Self Employment Income into Online Deluxe or Premier but if you have any expenses you will have to upgrade to the Self Employed version. Or any of the Desktop programs. But you will get the most help in the Home & Business version.

How to enter income from Self Employment
https://ttlc.intuit.com/community/self-employed/help/how-do-i-report-income-from-self-employment/00/26653

You might want to use Quicken or QuickBooks to keep tract of your income and expenses. There is one called QBSE QuickBooks Self Employed that works with Online Turbo Tax and will give you a free online tax return next year.
http://quickbooks.intuit.com/self-employed

You need to report all your income even if you don't get a 1099NEC or 1099Misc. You use your own records. You are considered self employed and have to fill out a schedule C for business income. You use your own name, address and ssn or business name and EIN if you have one. You should say you use the Cash Accounting Method and all income is At Risk.

After it asks if you received any 1099Misc it will ask if you had any income not reported on a 1099Misc. You should be keeping your own records. Just go through the interview and answer the questions. Then you will enter your expenses.

Self Employment tax (Scheduled SE) is automatically generated if a person has $400 or more of net profit from self-employment. You pay 15.3% SE tax on 92.35% of your Net Profit greater than $400. The 15.3% self employed SE Tax is to pay both the employer part and employee part of Social Security and Medicare. So you get social security credit for it when you retire. You do get to take off the 50% ER portion of the SE tax as an adjustment on 1040. The SE tax is already included in your tax due or reduced your refund. The SE tax is in addition to your regular income tax on the net profit.

Here is some IRS reading material……
IRS information on Self Employment
http://www.irs.gov/Businesses/Small-Businesses-&-Self-Employed/Self-Employed-Individuals-Tax-Center

Publication 334, Tax Guide for Small Business
http://www.irs.gov/pub/irs-pdf/p334.pdf

Publication 535 Business Expenses
http://www.irs.gov/pub/irs-pdf/p535.pdf