As the granddaughters of Queen Elizabeth, Princess Beatrice and Princess Eugenie are known as blood princesses. They received titles from their grandmother upon their birth but their future children won't, apparently, have the same royal distinctions. 

Princess Beatrice married Edoardo Mapelli Mozzi in July 2020, while Princess Eugenie married Jack Brooksbank in October 2018. Both their husbands have no royal titles and Queen Elizabeth did not bestow an earldom to the princess' husbands upon their wedding.

According to Hello, there has been no precedent for either man to receive royal titles.  Thus, the children of the York princesses may only inherit titles if Queen Elizabeth breaks tradition and makes their husbands earls. 

Upon their marriage, the princesses have also formally taken their husbands' names. Beatrice is legally referred to as Mrs. Edoardo Mapelli Mozzi while Eugenie is formally known as Mrs. Jack Brooksbank. 

Royal historian Carolyn Harris, also, said that royal titles in the U.K. are passed down to the male line only. So, a child born to a princess, but whose father is a commoner, won't have a title as well. 

Queen Elizabeth did grant an earldom to a member of the royal family by marriage. In 1960, Antony Armstrong-Jones married the monarch's sister, Princess Margaret, so their children were styled with royal distinctions -- David, 2nd Earl of Snowdon and Lady Sarah. 

In contrast, Queen Elizabeth offered to grant a peerage to her daughter, Princess Anne, and her husband Captain Mark Phillips when they married in 1979. However, the couple respectfully declined this gift so their children, Peter and Zara, do not have royal titles. 

Zara said, in an interview, that she feels "very lucky" she doesn't have a royal title because it took the pressure off, despite being raised in a royal household. She and her brother were able to grow up normally and get opportunities in their career, unlike their cousins Prince William and Prince Harry, who are constantly in the royal spotlight. 

Meanwhile, Princess Beatrice and Edo's children may still use a title, albeit not a British royal distinction. Edo belongs to a noble Italian family and his father is Count Alessandro Mapelli Mozzi. 

Edo is the lone male descendant of his father and will immediately inherit the title of count when the right time comes. Princess Beatrice will, also, be regarded as a countess in Italy, which will automatically make their future children count and countess in due time as well.