The two departures are not connected, but with parliament in summer recess, the timing of the departures gives Labour some breathing room to replace the pair during a quieter time in Westminster.
Fleming had been in the role for a year, according to LinkedIn. He took up the position of regional director (London) for the party this month.
According to LinkedIn, Fleming had been a northern regional comms officer at the party since February 2008, prior to taking the head of press position.
Ovenden has been with the party for three years, leading comms for London and the south east prior to his current lobby role. Before this, he had comms roles with charities Maggie's and the NSPCC.
He has taken a new job at M&C Saatchi, working on its Home Office counter extremism brief.
M&C Saatchi won the £60m creative brief, which includes community engagement as well as creating videos, websites and posters, in May last year. This is a separate project to the controversial Prevent programme, which is run by a separate part of the Home Office.
He will be working with Ros Trinick, who left the public affairs agency to lead comms on the account at the start of the year.