Alright, found a solution.
I've created a new default style for the ScrollViewer by following this MSDN article. Then I've changed the part where the scrollbars are placed.
Original behaviour
<Grid>
<Grid.ColumnDefinitions>
<ColumnDefinition Width="Auto"/>
<ColumnDefinition Width="*"/>
</Grid.ColumnDefinitions>
<Grid.RowDefinitions>
<RowDefinition Height="*"/>
<RowDefinition Height="Auto"/>
</Grid.RowDefinitions>
<Border Grid.Row="0" Grid.Column="1">
<ScrollContentPresenter CanContentScroll="True" Content="{TemplateBinding ScrollViewer.Content}" />
</Border>
<ScrollBar Orientation="Vertical" Grid.Row="0" Grid.Column="0" Minimum="0" Maximum="{TemplateBinding ScrollViewer.ScrollableHeight}" Value="{TemplateBinding ScrollViewer.VerticalOffset}" ViewportSize="{TemplateBinding ScrollViewer.ViewportHeight}" Name="PART_VerticalScrollBar" Visibility="{TemplateBinding ScrollViewer.ComputedVerticalScrollBarVisibility}" />
<ScrollBar Orientation="Horizontal" Grid.Row="1" Grid.Column="1" Minimum="0" Maximum="{TemplateBinding ScrollViewer.ScrollableWidth}" Value="{TemplateBinding ScrollViewer.HorizontalOffset}" ViewportSize="{TemplateBinding ScrollViewer.ViewportWidth}" Name="PART_HorizontalScrollBar" Visibility="{TemplateBinding ScrollViewer.ComputedHorizontalScrollBarVisibility}"/>
</Grid>
New behaviour
<Grid>
<!-- Presentation below is different from the default: the scrollbar is overlayed on the content. -->
<ScrollContentPresenter CanContentScroll="True" Content="{TemplateBinding ScrollViewer.Content}" />
<ScrollBar Orientation="Vertical" HorizontalAlignment="Right" Minimum="0" Maximum="{TemplateBinding ScrollViewer.ScrollableHeight}" Value="{TemplateBinding ScrollViewer.VerticalOffset}" ViewportSize="{TemplateBinding ScrollViewer.ViewportHeight}" Name="PART_VerticalScrollBar" Visibility="{TemplateBinding ScrollViewer.ComputedVerticalScrollBarVisibility}" />
<ScrollBar Orientation="Horizontal" VerticalAlignment="Bottom" Minimum="0" Maximum="{TemplateBinding ScrollViewer.ScrollableWidth}" Value="{TemplateBinding ScrollViewer.HorizontalOffset}" ViewportSize="{TemplateBinding ScrollViewer.ViewportWidth}" Name="PART_HorizontalScrollBar" Visibility="{TemplateBinding ScrollViewer.ComputedHorizontalScrollBarVisibility}"/>
</Grid>
Now the scrollbars are overlayed on the content so they don't take additional space when visible. Of course the content should now reserve space for the content.