Let's 1st understand what these planets represent on their own?

Sun - Sun represents Soul, Personality, King, Government, Father, Authority, Career, Top Positions, Ego, Self Esteem, Health, Education etc.

Saturn - It is all that which we don't like in our life. Saturn is Delay, Frustration, Restriction, Limitation and Anxiety. It is people of Authority, like Boss at your Work Place or Father-Figure at Home. Saturn is our Karmic Debt from past-life. Saturn is the things we didn't treat too well in our past life. Saturn is our Challenges and Lessons of life. Saturn is Old-Age, Teeth/Bones in Human Body, Diseases, Changes etc. Saturn represents our Step-Relations, it means that wherever Saturn sits, people related with that house give us Step Treatment. Saturn is Hard Work, Efforts and most importantly Perseverance etc. Nothing comes easy and in one go with Saturn. You have to work hard continuously over the longest possible period of life. Saturn makes a person Legend as Legends have one thing in common that they choose one field and work hard in it for whole life. 

House Ruler - Besides these karaka things, Sun & Saturn will also rule some houses in your chart and will represent things related to that house. Of course we can't cover conjunctions Ascendant and house wise else it will never be over. So, you have to include those things at the time of interpreting this conjunction. Also, we need to keep in mind all Yogas and Raj-Yogas which their conjunction would be making to completely understand the conjunction. Here, we can only cover meaning as per karakas. 

Meaning of Sun-Saturn Conjunction - So, this is one of the most stressful conjunction you can get. No matter how much positive we want to think and how much we want to console ourselves, this conjunction will remain the most stressful as two of the biggest enemies are together. It is like real war zone in the house they are together. First of all,it can give real distant relation from Father or Authorities. It shows real stress with Father. It can also be the case that person always stayed away from Father. Authorities and people in power may never treat this person well. This person will grow in life within an environment of certain fear because Confidence (Sun) is deeply hurt due to Fear (Saturn). No matter how far or near they are in degrees, this fear factor will remain. Closer the degree between two, more the fear. Positively, they can grow as a very responsible person for the simple reason that Father was always away and they had to take up all the responsibilities early in life. This conjunction can help someone in Legal or Govt field. They are highly disciplined and law abiding. Give them a rule to follow and they can follow that rule whole of life. In whichever house this conjunction is taking place, that house will give some good results only in 30s. Till then, it is all hard work. Person's Career growth may be delayed and he may realize his true self only in 30s. They can have stage-phobia. As I said, closer Sun-Saturn are, bigger the stress and fear. If Saturn is combust, person can have real life-long issues with authorities and law. Overall, this is a conjunction that helps in Law and Govt jobs, but even there nothing will come easily. Person has to be hard working and law abiding through out. If he is not law abiding, then he cannot expect Sun-Saturn to do any favor. Then he may have a life-long date with Legal Agencies. 

Who will win the Conjunction? - None or Both. Although Sun is Father and has all the fire but Saturn's impact on Sun will be equally visible. Even if Saturn is combust, it will only make Saturn further agitated. There mutual tussle of over-powering each other will result in nothing but stress, delay, fear and frustration. Only way to deal with this conjunction is to put all your hard work and leave the results on God. 

Dignity & Strength - 

This conjunction will remain stressful in any house and in any sign. Things related with that house and sign should be expected in results only after immense hard work and after mid-30s. In signs like Aries, Cancer, Leo, Sagittarius and Pisces, Sun can have little more say but Saturn will have its malefic impact also on the things related to that sign/house. In signs like Taurus, Gemini, Virgo, Libra, Capricorn and Aquarius, Saturn will be more powerful but that only means that person will struggle with Father, Career and Authority as Sun is weak. Scorpio is not good for either. A real challenging situation.