Economically it doesn't make much difference, but I like to control the account because I can adjust the amount I put aide each month based on the new tax rates that come out each spring. This allow me finer control. I also know that the bills have been paid, I had one lender years ago that failed to pay the property tax bill, I had to end the money in to the county, and then pend months fighting the lender to get the money back. Now I avoid escrow accounts.
The money being collected by the mortgage servicing company for property taxes and property insurance goes into a separate account. The company insists on handling the funds to make sure that these bills are paid on time, thus protecting their investment. The failure to pay the taxes leaves the property subject to forfeiture via tax lien. The failure to pay property insurance leaves the house unprotected if there is a fire or other incident.
You can avoid the use of an escrow account if you have enough equity in the account. Some lenders ask for you to provide proof of payment each year if you are going to pay it yourself.
At the end of each year the servicing company will provide you with an accounting for interest on the loan, and the amount of money spent on taxes and insurance. Also expect that they will make adjustments to the monthly withholding based on estimated increases for taxes and insurance.
Depending on your financial situation the interest, taxes may be included on Schedule A. If you have a rental property the interest, taxes and insurance are considered expenses that you can write off.
The biggest issue with escrow accounts is that the company can have a buffer built in to protect them from unexpected increases. Many people view the calculation of the buffer a confusing and feel that they are overpaying.
If you want to avoid the escrow account you should make sure that each month you put the money into a separate account so that when the property tax bill is due you can pay it on time. When savings accounts earned significant interest it was possible to make a little money in the deal, but that hasn't been true for the last few years.