Previous post makes good points - as a massage therapist I'll add a couple more thoughts.
It is hard to say for individual cases as it depends on the state of your chest infection and impact on other systems. But in general if you are beyond the stage of first infection and have no fever or other contraindications for massage then it MAY help as you suggest.
Moderated pressure can be explored within the session, finding the level/depth you find useful and tolerate. Everyone is different regardless of the infection - you may find superficial layers or lymphatic drainage enough to cope with or may find it 'ticklish' and unpleasant compared to a medium pressure, or simply unhelpful to clear the mucus.
Strongly recommend you seek a massage therapist with either experience in remedial massage (often taught alongside sports massage but don't let that put you off!) or other health related experience/training rather than a beauty/hair salon trained masseuse as their training can be limited to fixed routines aimed at relaxation and you'd benefit from a more tailored session.
Assuming your therapist gives you the ok to treat, perhaps have a big box of tissues as if it works you'll be coughing up a fair bit and it's best to be prepared!
I wouldn't recommend you lie on your back if it still prompts coughing, lying face down is probably easiest and is what would be used for example with someone with cystic fibrosis whose lungs need regularly cleared. But in the first instance given the limited info I have from a post compared to a full assessment and forms, I wonder if you might find a seated massage on an onsite massage chair more comfortable, which supports your front and legs but gives good access to the shoulders and upper back.
Advantage of that option is that you generally remain clothed and you may need to keep warm and wrapped up while your condition clears. Your lungs are under less pressure than with your bodyweight full on them, and you can move to a table as and when it seems appropriate. As I say, CS sufferers get lungs cleared lying prone but it is quite a hard process and you aren't necessarily going to benefit from that level of intervention as previous poster already noted.
I am assuming you have either had necessary treatment to address any continuing bacterial infection and/or excluded infectious viral conditions that might pass on.. A therapist won't thank you if they can't work for several weeks, we're mostly self employed and no income is hard, but you can't sneeze and cough all over your clients every hour of the day!