Here's a very simple solution. Let's start with a graphic example and reference.
We'll say the green line is your 2007 line and the red line is 2010. Are they both in the same shapefile/feature class? Regardless, you'll want them both added and it would be helpful if they were symbolized differently so you can tell them apart by looking. If in the same featureclass I would assume they have a year attribute you could symbolize on.
The first step is to connect the matching endpoints of the two lines (blue lines in the graphic). If the two lines already share endpoints, this won't be necessary, but note they do need to be snapped together with no gaps. Whether you do this in one of/the line class or a new one is up to you.
Then you need to create polygons out of the enclosed areas (magenta Xs) between the two lines. You'll need an editable polygon feature class to do this. It should have at least two attributes - area (double) and type (text). To convert those enclosed areas to polygons you can use the Feature to Polygon tool or construct polygons as noted in this question.
Now you need to differentiate whether an area is a 'gain' or 'loss'. Open the polygon attribute table and start entering one of those two values in for each shape in the appropriate field. From left to right in the example, the first is a gain, the next a loss, then a gain, and so on. This will work if you're not looking at a large area or there aren't a tremendous number of crisscrosses. Otherwise, see alternative below. To get the areas, right-click the Area attribute column header and choose Calculate Geometry, pick area and your desired units, making sure you either have nothing selected or uncheck the box that says only calculate selected records.
Finally you need your summary. If only a few shapes, you might sort the table and do the math by hand or selecting one type or the other and doing a right-click > Statistics (which should only give you the result for selected records). If a lot, you can use the Summary Statistics tool with a CASE field of 'type' (gain or loss) and a statistic field of Area with the SUM type. That will generate a table with the sum of gain and loss. Add the two together and you get net gain or loss.
Now, if your polylines are very large or there are a lot of crossings, this won't be very efficient because you have to identify if an area is gain or loss. The alternative is to turn your lines themselves into polygons, meaning you will have to draw a line from one end of the green line down and then over to the other, doing the same for the red line. Note that the parts you draw should be identical/snapped. When you create polygons out of those enclosed areas, you'll have a 2007 and a 2010 polygon. Note this is different than above where you're just creating polygons out of the areas of overlap.
Once you have two polygons, you'll run Union on them. The resulting shapefile/feature class will cut up the two polygons wherever they cross with attributes that tell you if the new shape was from one, the other, or both of the two inputs. You should have one big area that is both 2007 and 2010, representing no change. Then all your areas where the lines overlap in the first method will be either 2007 or 2010, but not both. Anything that is a 2007 is a loss since there's nothing overlapping in 2010, and vice versa. Once again you can create an Area attribute and calculate the areas in your desired units, and using either Summarize in the attribute table or Summary Statistics you should be able to get the total for each area type (though you may have to do a little attribute manipulation to simplify things).